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Rational Function Systems
and Electrical Networks
with Multi-Parameters

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NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
World Scientific
Rational Function Systems
and Electrical Networks
with Multi-Parameters
Kai-Sheng Lu
Wuhan University of Technology, China

Kai-Sheng Lu
Wuhan University of Technology
Wuhan, 430063, China
[email protected]

Copyright  2012 by
Higher Education Press
4 Dewai Dajie, Beijing 100120, P. R. China and
World Scientific Publishing Co Pte Ltd
5 Toh Tuch Link, Singapore 596224

All rights reserved. No part of this book may be reproduced or transmitted in any
form or by any means, electronic or mechanical, including photocopying,
recording or by any information storage and retrieval system, without permission
in writing from the Publisher.

ISBN-13: 978-981-4412-42-1

Printed in P. R. China

Preface
The doctoral course “Rational Function Systems and Electrical Networks
with Multi-parameters” has been delivered by the author, for many years
based on his extensive research. With a lack of corresponding books in this
ﬁeld, there is a challenge for higher education to have a framework and text to
work from. It is quite necessary to transform the course lecture notes and the
knowledge acquired over a long careerfocusedinthisareaintoasystematic,
comprehensive book to assist in teaching and research.
Due to the inconvenience in exploring structural properties using linear
system theory and electrical network theory over the real ﬁeldR, the author
uses the matrices over the ﬁeldF(z) of rational functions in multi-parameters
to describe coeﬃcient matrices of systems and electrical networks and inves-
tigates their structural properties based on the description of systems and
electrical networks overF(z). The bookRational Function Systems and Elec-
trical Networks with Multi-parametersis divided into ﬁve chapters. Chapter
1 introduces the background and meaningof systems and electrical networks
overF(z). In chapter 2, Matrix theory is extended to the ﬁeldF(z); the
reducibility condition of a class of matrix and its polynomial overF(z)is
discussed in detail; the deﬁnition of type-1 matrix and two basic properties
is given; the fact that type-1 matrix is of the two properties is proved and
the variable replacement condition for independent parameters is introduced.
Chapter 3 explores the structural controllability and observability of linear
systems overF(z); and introduces some new conclusions in time domain and
frequency domain. Chapter 4 shows thestructural properties of electrical
networks overF(z): the separability, reducibility, controllability, and observ-
ability of RLC networks overF(z) and structural conditions of controllabil-
ity and observability overF(z); the separability, reducibility, controllability,
and observability of RLCM networks overF(z); the state equation existence
condition of active networks overF(z) and controllability and observability
conditions overF(z). Chapter 5 describes further thoughts on the ﬁeld.
Since the object of research in this book is Rational Function Systems and
Electrical Networks with Multi-parameters, the results obtained are usually
clear and intuitive, and are convenient to analyze and design systems and
electric networks. For instance, there is such a conclusion that for an RLC

vi Preface
network overF(z) (this means in this kind of RLC networks all the resis-
tors, capacitors, and inductances are regarded as independent parameters)
without sources, if it has no all-capacitor cut-sets or all-inductor loops and
it is inseparable (these are all structural conditions), then the characteris-
tic polynomial of this network is irreducible overF(z) (it does not need to
calculate the characteristic polynomial, and we just need to observe the net-
work structure). The reducibility of the characteristic polynomial has a direct
relationship with controllability, observability, and stability. So it is easy to
analyze and design an RLC network whose characteristic polynomial is irre-
ducible according to this conclusion. The description of systems and electrical
networks overF(z) is a useful tool to investigate the structural properties of
systems and electrical networks. Furthermore, the real ﬁeld Ris the subﬁeld
ofF(z), so the conclusions overF(z) are more general than those overR.
This book summarizes the author’s research achievements over the past
20 years with four projects for National Nature Science Foundation of China
(subjects include:Research on Electrical Network Theory overF(z)and
Computer Assistant Analysis, 1995 – 1997;Researching Electrical Network
Structural Properties Using Matrix overF(z), 2002 – 2004;Research on Struc-
tural Controllability and Observability of Systems overF(z), 2006 – 2008;Re-
search on Separability, Reducibility, Controllability, Observability and Stabil-
ity of Active Networks overF(z), 2010 – 2012) and two projects for the Nature
Science Foundation of Hubei Province of China, which is at the leading edge
of scientiﬁc research.
The description of systems, electrical networks, and matrices overF(z)
in this book is diﬀerent from other relevant books, which introduce linear
systems, electrical networks and matrices overR.Inthisbookweexploresys-
tems, electrical networks, and matrices overF(z). This book involves three
subject areas: systems, networks and matrices overF(z),whichisanachieve-
ment of interdisciplinary research. Sothere is a close connection among the
three subjects in this book. For example, the reducibility condition of a class
of rational function matrix introduced in Chapter 2 is the important base of
Chapter 3 and Chapter 4. Usually systems, electrical networks, and matrices
are introduced by three distinct subject area books, which are independent
to each other.
This book can be used by postgraduate students, Ph.D students, college
teachers, researchers and engineers in the ﬁeld of electronic and electrical
engineering, automatic control, and applied mathematics matrix theory .
If any deﬁciency or mistake should appear in the book due to my over-
sig
ht, I welcome your comments and feedback to improve future editions and
expanding the journey of scientiﬁc research.
Kai-Sheng Lu

Acknowledgments
This book is dedicated to my mother Gong-Gao Liu. She entered Tsinghua
University in 1938 and graduated from Southwest Union University, which
consisted of Beijing University, Tsinghua University and Nankai University
during the period of World War Two, and was an assistant of Mr. Yi-Duo
Wen at the Institute of Arts Research in Tsinghua University. I am sad to say
my mother lost her husband in a car accident at the age of 31. Although her
schoolmate fell in love with her, she put her children’s needs above all and
stayed a widow. She sacriﬁced her happiness and responded to all the duties
of bringing us up despite all kinds of hardships. Without her instruction,
ediﬁcation, and encouragement, I could not have completed this book in my
poor health.
The research achievements in this book were supported by the National
Nature Science Foundation (NNSF)mentioned aboveunder grant: 69472008,
50977069, 60574012 and 50177024. I would like to oﬀer my sincere gratitude
to the NNSF committee.
I would also like to extend my thanks to my students: Xiao-Yu Feng,
Guo-Zhang Gao, Ke Zhang, Zong-Tao Wang, Guan-Min Liu, Qiang Ma and
Hu-Ping Xu who have taken part in the research works of the NNSF; Qiang
Ma and Yu-Peng Yuan, especially Qiang Ma, who have been devoted to the
tedious work of editing the e-document of the book.

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Contents
Preface·············································· v
Acknowledgments····································· vii
1 Introduction······································· 1
References········································· 8
2 Matrices over FieldF(z) of Rational Functions in
Multi-parameters··································· 11
2.1 Polynomials over FieldF(z)orRingF(z)[λ]············· 11
2.2 Operations and Determinant of Matrix overF(z)·········12
2.3 Elementary Operations of Matrices overF(z) and Some
Conclusions····································· 12
2.4 Operation and Canonical Form of Matrix overF(z)········14
2.4.1 Matrix overF(z) and its canonical expression·······14
2.4.2 Characteristic matrix························· 23
2.4.3 Two canonical forms of nonderogatory matrix·······27
2.4.4 Rational canonical form and general Jordan
canonical form······························ 32
2.5 Reducibility of Square Matrix overF(z)················ 35
2.6 Reducibility Condition of Class of Matrices overF(z)······36
2.6.1 A class of RFM····························· 36
2.6.2 Some lemmas and deﬁnitions··················· 36
2.6.3 Reducibility condition························ 39
2.6.4 Applications······························· 48
2.6.5 Summary································· 51
2.7 Two Properties·
································· 51
2.7.1 Some lemmas······························ 52
2.7.2 Type-1 matrix has two properties················ 54
2.7.3 Problems································· 59

x Contents
2.8 Independent Parameters and a Class of Irreducible
Polynomials overF(z)[λ]··························· 59
2.9 Conclusions····································· 64
2.10 New Model and Its Reducibility······················ 68
2.10.1 The new model···························· 68
2.10.2 Reducibility condition······················· 69
References········································· 71
3 Controllability and Observability of Linear
Systems overF(z)·································· 73
3.1 Controllability and Observability in Time Domain·········73
3.1.1 Preliminaries [Lu, 2001]······················· 73
3.1.2 Controllability criteria [Lu, 2001]················ 75
3.1.3 The canonical decomposition of controllability and
observability of systems······················ 87
3.1.4 Criterions to linear physical systems·············· 89
3.1.5 Applications to control systems·················100
3.2 Controllability and Observability in Frequency Domain·····102
3.2.1 General systems [Lu et al., 1991]················102
3.2.2 SC-SO of composite systems [Lu et al., 1991]·······106
3.2.3 Polynomial matrix [Liu, 2008]·················· 113
References········································· 120
4 Electrical Networks overF(z)························ 121
4.1 Resistor-Source Networks overF(z)··················· 121
4.1.1 Introduction······························· 121
4.1.2 General resistor-source networks·················123
4.1.3 Unhinged networks·························· 131
4.1.4 Eﬀects of single source························ 135
4.2 Separability and Reducibility Conditions of RLC Networks
overF(z) and Their Applications····················· 140
4.
2.1 Introduction······························· 140
4.2.2 Preliminaries······························· 141
4.2.3 Separability condition························ 143
4.2.4 Separability and reducibility···················· 146
4.2.5 Applications······························· 147
4.3 Controllability and Observability of RLC Networks
overF(z)······································ 149

Contents xi
4.4 Structural Condition of Controllability for RLC Networks
overF(z)······································ 151
4.4.1 Separability conditions························ 151
4.4.2 Structural controllability conditions··············154
4.5 Structural Condition of Observability for RLC Networks
overF(z)······································ 155
4.5.1 Node voltage equation and two results············156
4.5.2 Structural condition of observability overF(z)······158
4.6 Separability, Reducibility, Controllability and Observability of
RLCM Networks overF(z)························· 160
4.6.1 Preliminaries······························· 160
4.6.2 Separability································ 161
4.6.3 Reducibility······························· 166
4.6.4 Controllability and observability·················168
4.6.5 Structural condition of controllability and observability
overF(z)································· 173
4.7 Existence of State Equations of Linear Active Networks
overF(z)······································ 177
4.7.1 Existence condition of state equation overF(z)······177
4.7.2 Application································ 181
4.8 A Suﬃcient Condition on Controllability and Observability of
Active Networks overF(z)·························· 185
4.8.1 Preliminaries······························· 186
4.8.2 Suﬃcient condition of controllability overF(z)······189
4.8.3 Applications······························· 189
4.9 Conditions onB11ﬃ=0and
ﬃ
Cﬃ= 0 of Active Network
overF(z) and Reducibility Condition ofAand Their
Applications to Controllability and Observability·········190
4.9.1 Preliminaries······························· 191
4.9.2 Partitioning ofyandu 2fromu 1················193
4.9.3 Conditions ofB11ﬃ=0························ 198
4.9.4 Reducibility ofAand conditions of
ﬃ
Cﬃ=0··········203
4.9.5 Examples································· 207
4.9.6 Applications to controllability and observability
overF(z)································· 213
4.9.7 Method of designing a structural controllable and
observable active network with normal form········223
4.10 Computer Assistant Analysis Program for Networks
overF(z)······································ 224

xii Contents
4.10.1 Software interface illumination·················224
4.10.2 Structural analysis process description of the
software·································· 226
4.10.3 Software functions·························· 232
References········································· 235
5 Further Thought ··································· 237
5.1 Independent Parameters — The Third Type of Variables
of Systems····································· 237
5.2 Physical Realization······························· 238
5.2.1 Canonical state space description of linear
time-invariant systems························ 238
5.2.2 Two basic properties························· 239
5.3 Some Issues····································· 240
5.3.1 Is it irreducible when exist interaction?············241
5.3.2 Dimension of nonzero modeﬃnumber of
independent parameters······················· 242
5.3.3 Design of active networks being SC-SO and stable····242
5.4 Quasi Structural Controllability and Observability Concept of
Nonlinear Systems and Its Applications················243
5.4.1 Preliminaries······························· 243
5.4.2 Quasi-structural controllability of nonlinear systems··245
5.4.3 Applications······························· 246
5.4.4 Conclusions································ 249
References········································· 249
Appendix············································ 251
Appendix A Some Well-known Results···················· 251
A.1 Linear systems theory overR···················· 251
A.1.1 Controllability criterions in time domain······251
A.1.2 Polynomial matrix theory in frequency domain·253
A.2 Graph theory······························· 258
A.3 Linear graph································ 262
A.3.1 Linear graph·························· 262
A.3.2 Relationships in RLC networks·············272
Appendix B Some Relevant Proofs in Theorem 4.11··········274
B.1 The proof of
Δ
C
12ﬃ=0·························· 274
B.2 The proof of
Δ
Y
12ﬃ=0·························· 275
B.3 The proof of
Δ
G
12ﬃ=0·························· 277

Contents xiii
B.4 The proof of
Δ
L 12ﬃ=0or
Δ
Z 12ﬃ=0·················· 279
Appendix C Proof of Theorem 4.15······················ 282
Appendix D Proofs of some conclusions··················· 287
D.1 The proof of Theorem 4.18····················· 287
D.2 The proof of Theorem 4.19····················· 293
D.3 Some results································ 297
References········································· 302
Index··············································· 303

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1 Introduction
AmatrixA=(a ij)n×mis called the matrix over the real ﬁeld or real number
matrix, if each entrya
ijof this matrix is a real number. A system
˙
X=
AX+BU, Y=CX+DUis called a linear system (including an electrical
network, or a network for short) over the real ﬁeld or a real number system,
if its coeﬃcient matricesA,B,C,andDare real number matrices.
We know that linear systems and networks over the real ﬁeld are explored
very well and are successfully used for analysis and design of systems and net-
works[Chen, 1984
ﬃZheng, 2002ﬃChen et al., 1992ﬃChen, 1976; Balabanian,
et al., 1969; Chen, 1987]. However, people ﬁnd that real number matrices
are not convenient to analyze structural properties of physical systems (e.g.,
structural controllability and structural observability). Take the liquid level
control system shown in Fig. 1.1 as an example.
Fig. 1.1Liquid level control system
In Fig. 1.1,G 1(s)=
Δq
i(s)
E(s)
=K
p
Δ
1+
1
Tis
Σ
,G
2(s)=
Δh(s)
Δqi(s)
=
K
Ts+1
.
The closed loop transfer function of this system is
Δh(s)
Δhi(s)
=
b
1s+b 0
s
2
+a1s+a 0
.
The block diagram representation of the system can be denoted by
wherea 0=
KK
p
TiT
,a
1=
KK
p+1
T
,b
0=
KK
p
TiT
,b
1=
KK
p
T
.
Then the state equation of the feedback system can be denoted by
˙
X=AX+BΔh
i,Δh=CX,
whereX=(ω,˙ω)
T
,ωis the output signal of the left block, ˙ωis a derivative

2 1 Introduction
ofω,and(ω,˙ω)
T
denotes the transpose of (ω,˙ω),
A=
⎛
⎜
⎝
01
−
KK
p
TTi
−
1+KK
p
T
⎞
⎟
⎠,B=

0
1

,C=
Δ
KK
p
TTi
KKp
T
Σ
.
(1.1)
For this system with given structure,only when all physical parameters
K
p,Ti,K,andTtake values,A,B,andCare real number matrices and the
system is real number system. So analysis results of this real number system
(such as the controllability of (A,B), observability of (A, C
T
), reducibility
of characteristic polynomial det (λI−A), and so on) are determined by two
factors: physical system structure and parameter values. But what is the
individual eﬀect of system structure is not distinguishable.
To explore the eﬀect of system structures, various parametrizations were
appeared:
Paper [Lin, 1974] proposed a structured matrix (SM), whose entry is
either constant zero or independent nonzero, that is, the nonzero entries are
independent parameters. For example,
A=

z
10
z
2z3

,
where the three nonzero entriesz
1,z2,andz 3are independent parameters.
Papers [Shields et al., 1976; Glover et al., 1976; Davison, 1977; Hosoe et
al., 1979; Mayeda, 1981; Li et al., 1996] used SM investigating structural
controllability of MIMO systems.
Papers [Corpmat et al., 1976; Anderson et al., 1982; Willems, 1986] in-
troduced one-degree polynomial matrix, whose entries are one-degree poly-
nomials in independent parameters. Such as
A=

11
00

+z
1

11
10

+z
2

10
01

=

z
1+z2+1z 1+1
z
1 z2

is a one-degree polynomial matrix in the independent parametersz
1andz 2.
A matrix is called a column – structured matrix (CSM) if the diﬀerent
entries in a column of the matrix contain the same parameter factor but the
factors in diﬀerent columns are independent of each other [Yamada et al.,
1985]. For example,
A=

3z
1z2
2z14z2

,
wherez
1andz 2are parameter factors in the ﬁrst column and second column,
respectively, and they are independent.

1 Introduction 3
A matrix of the formM=Q+Tis said to be a mixed matrix if the
nonzero entries ofTare algebraically independent over the ﬁeldKto which
the entries ofQbelong [Murota, 1987, 1989a, 1989b, 1993, 1998]. Take the
following matrix as example,
A=Q+T=

0
√
2
30

+

z 10
0z
2

,
whereQis a matrix over the real ﬁeld and the nonzero entriesz
1andz 2in
Tare algebraic independent over the real ﬁeld, that is, the real ﬁeld does not
contain the two members.Ais called a mixed matrix, whereKis the real
ﬁeld.
Clearly, the inverse matrix of the full rank square SM, CSM, one-degree
polynomial matrix, or mixed matrix is generally not SM, CSM, one-degree
polynomial matrix, or mixed matrix. To overcome this problem, papers [Lu
et al., 1991, 1994] introduced rational function matrix in multi-parameters
(RFM) to describe the coeﬃcient matrices of systems and networks and based
the description of systems and networks on RFM to research their structural
properties.
Letz
1,...,zqdenoteqindependent parameters (can also be variables or
indeterminates), not constants or numerical values. Letz=(z
1,...,zq)∈R
q
,
R
q
is the domain of deﬁnition forz, and it can also be called parameter
space. LetF(z) denote the ﬁeld of all rational functions with real coeﬃcients
inqparametersz
1,...,zq,andF(z)[λ] denote the ring of allF(z)-coeﬃcient
polynomials inλ. Such as
√
5z1+2z 2+3
4z1+z3
∈F(z),
where
√
5,2,3,4, and 1 are real coeﬃcients; because polynomial ring inzis
a subset ofF(z),
√
5z1+2z 2+3∈F(z); because the real ﬁeld is a sub-ﬁeld
ofF(z), any real number is a member ofF(z). It is clear that
(5z
1+z
3
2
)λ
2
+
√
6z6λ+
8z
2
3
+1
3z4z5+z2z6+
√
2
∈F(z)[λ].
Deﬁnition 1.1If any entry of matrixMis a member ofF(z) (i.e., rational
function inz
1,...,zq), then matrixMis called a rational function matrix
(RFM) inzor a matrix overF(z); if the coeﬃcient matrices of a system
(including network) are considered to be RFMs, the system is called the
rational function system in multi-parameters, simply called rational function
system (RFS), or system overF(z).
Obviously, the inverse matrix of a full rank square RFM is also an RFM.

4 1 Introduction
Papers [Lin, 1974; Shields et al., 1976; Glover et al., 1976; Davison, 1977;
Hosoe et al., 1979; Mayeda, 1981; Li et al., 1996; Corpmat et al., 1976;
Anderson et al., 1982; Willems, 1986; Yamada et al., 1985; Murota, 1987,
1989a, 1989b, 1993, 1998] are of mathematical signiﬁcance. However usually
these matrices deﬁned in the papers can not describe the physical systems
completely, so they are not directly usedto explore the structural properties
of physical systems. Take Fig. 1.1 as illustration. The independent param-
eters of this system should be the 4 physical parametersK
p,Ti,K,andT.
From Eq. (1.1) and the above deﬁnitions we know that matrix over the real
ﬁeld, SM, one-degree polynomial matrix in independent parameters, CSM
and mixed matrix can not describe the matricesA,B,andCcompletely, but
all these three matrices are RFMs and the system (A, B, C)isRFS,where
z=(K
p,Ti,K,T). Why RFM can describe the structure of physical systems
is that the conception of RFM is more general, and matrix over the real ﬁeld,
SM, one-degree polynomial matrix, CSM and mixed matrix can be treated
as special RFMs. So the research work of RFS has both mathematical and
physical signiﬁcance.
It should be emphasized that the conclusions overF(z) is determined
by structures of systems and networks only, not the values ofz, because
parameters are not evaluated with regard to conclusions overF(z), which
eliminates the value eﬀect and only leaves the structure eﬀect. Let us see the
following conclusions overF(z):
1) If an RLC network overF(z) without sources has no all-capacitor cut-
set, or all-inductor loop and is unhinged
1ﬃ
(this is structural condition), then
the characteristic polynomial det (λI−A) of the network is an irreducible
polynomial overF(z)[λ] (without calculating characteristic polynomial, and
only observing the structure of this network) [Lu et al., 1998].
2) If an unhinged RLC network with sources overF(z) has no all-voltage
source and capacitor loop, or all-current source and inductor cut-set, and if
its network without sources (let voltage sources be short circuited and current
sources be open circuited) is unhinged and has no all-capacitor cut-set, or
all-inductor loop, then this network with sources is controllable overF(z)
[Lu, 2003].
3) If an RLC network without sources overF(z) is unhinged and neither
all-capacitor network nor all-inductor network, then the network is observable
overF(z) when any network variables (node voltages and/or branch currents)
are outputs [Lu et al., 2005b]; where all resistors, capacitors and inductors
of the network are treated as independent parametersz.
1ﬃAn RLC network is said to be separable (or hinged) if the network has at least one
sub-network which has at most one node in common with its complement sub-network,
otherwise it is an unhinged network.

1 Introduction 5
For example, consider the network, as shown in Fig. 1.2, wherez=
(C, L, R
1,...,R4). Clearly, its network without sources (let the voltage source
be short circuited) is unhinged and has no all-capacitor cut-set, or all-inductor
loop. So from conclusion 1) we know that the characteristic polynomial
det (λI−A) of this network is an irreducible polynomial in ringF(z)[λ].
From Fig. 1.2, we know that there is a source in this unhinged network and
no all-voltage source and capacitor loop, or all-current source and inductor
cut-set; its network without sources is unhinged and has no all-capacitor
cut-set, or all-inductor loop. So from conclusion 2), this network is control-
lable overF(z). If take the node voltageu
R1of resistorR 1as output in its
network without sources, then for the network without sources is unhinged
and is not all-capacitor or all-inductor network, the network is observable
overF(z) from conclusion 3). Application of these conclusions only needs to
observe network structure. So conclusions overF(z) are only determined by
structure, and the properties of systems overF(z) or simply called properties
overF(z) (such as controllability and observability overF(z)) are equivalent
to structural properties (such as structural controllability and observability).
The description of systems overF(z) is a useful tool to explore structural
properties.
Fig. 1.2An RLC network
We make some primary investigation to structural properties of RFSs us-
ing this tool: Research on the reducibility for a class of square RFM and its characteristic polynomial; controllability and observability overF(z); sepa-
rability, reducibility, controllability, and observability of networks overF(z).
Some obtained conclusions will be introduced in Chapters 2 – 5.
Now we will show the diﬀerences to the existing well known theories. Lin-
ear system theory can be divided into four parallel embranchments: state- space method, frequency domain method, geometric theory, and algebra the- ory according to mathematic tool and system description [Zheng, 2002].
The basic description of state-space (time domain) method is
˙x=Ax+Bu, y=Cx+Du,
wherex∈R
n
,u∈R
m
,y∈R
p
,A,B,C,andDaren×n, n×m, p×n
andp×mreal number matrices respectively. This is a linear time-invariable

6 1 Introduction
system, which is a real number system above (the term “real number system”
in this book is to distinguish to RFS). In this book,A,B,C,andDare
considered to bematrices overF(z), and systems are treated to be RFSs,
and structural properties dependingon system structures not on parameter
values are researched.
Frequency domain method is based on the state-space method. The trans-
fer function matrixC(sI−A)
−1
B+Dof the real number system is a matrix
over the ﬁeldF(s) of rational functions in only one complex variables,where
F(s) denotes the ﬁeld of all rational functions with real coeﬃcients ins,that
is, the coeﬃcients ofsare real numbers. However although the entry of the
transfer function matrixC(sI−A)
−1
B+Dof RFS is rational function ins,
the coeﬃcients ofsare members overF(z), which are rational functions in
z
1,...,zqand not just real numbers — real numbers are a special situation.
Or we can say that this transfer function matrix is a matrix overF(z;s),
whereF(z;s) denotes the ﬁeld of all rational functions with real coeﬃcients
inq+ 1 independent parameters or indeterminatesz
1,...,zq,s.
The algebra theory of linear systems(see Chapter 10 of [Kalman et al.,
1969]) explores the linear systems over arbitrary and certain ﬁeldKof num-
bers — generally it is the real ﬁeldR, where the state vectorx∈K
n
, input
vectoru∈K
m
, output vectory∈K
p
, and the coeﬃcient matricesF,G,and
H(now customarily denoted byA,B,andC)aren×n, n×mandp×n
matrices respectively overK.SinceKis the ﬁeld of numbers, the description
of linear systems overKis not easy to express and explore the structural
properties of physical systems.
The geometric method of linear system [Wonham, 1979] explores linear
space over the real ﬁeld and the ﬁeld of complex numbers, other thanF(z),
so it is not easy to describe the structure of physical systems yet.
In this book, we utilize the generally used and eﬀective method of the
state-space and frequency domain, and try to extend the theory of linear
time-invariable systems (real number systems) to RFSs. The signiﬁcance of
this is as follows:
First, description overF(z) is a useful tool to explore the structural prop-
erties of systems and networks. Research of systems and networks overF(z)
is a work of mathematic and physical signiﬁcance.
Furthermore, researching and analyzing properties of systems and net-
works overF(z) (structural properties) has more practicable meaning than
that over the real ﬁeld. For instance, the controllability and observability of
system are as shown in Fig. 1.1. WhenK
p=2,T i=1,K=3,T=1,or
denoted byz=(K
p,Ti,K,T)=
z=(2,1,3,1), substituting into Eq. (1.1)
yields

1 Introduction 7
A=

01
−6−7

,B=

0
1

,C=
γ
66
ρ
. (1.2)
According to theory of real number systems, clearly this system (1.2) is not
observable. But the determinant of observability matrix for RFS
det (C
T
,A
T
C
T
)
T
=
K
2
P
K
2
(T−T i)
T
2
i
T
3
is a nonzero member overF(z), and the system is observable overF(z), that
is, structurally observable. The points (such as
z=(2,1,3,1)) which make
this system unobservable overRform a set{z∈R
4
|det (C
T
,A
T
C
T
)
T
=0},
which is only a hypersurface in parameter spaceR
4
. BecauseK p,Ti,K,T
are physical parameters, it is impossible that their taken values are accurate enough to make det (C
T
,A
T
C
T
)
T
= 0. As for the real number system (1.2)
is unobservable, it can be understood that the man-made or mathematical constrainz=
z=(2,1,3,1) of the parameters makesT−T i=0,thatis,
det (C
T
,A
T
C
T
)
T
= 0. To a control system (physical system), if it is control-
lable and observable overF(z) (structural controllable and observable), then
it is always controllable and observable over the real ﬁeld, or in the view of
robust, it is robust controllable and observable (it can be treated as a system
with certain structure and with uncertain parameters) for almost allz∈R
q
.
So researching and analyzing properties of systems and networks overF(z)
(structural properties) has more practicable meaning than that over the real
ﬁeld.
Moreover, it is of theoretical value to research the structural properties
of systems overF(z). The real ﬁeld is a subﬁeld ofF(z); and SM, CSM,
one degree polynomial matrix and mixed matrix can be considered to be a
kind of matrices overF(z). So the criterions obtained overF(z)canbemore
generally used. For example, there is a conclusion overF(z) in [Lu, 2001]:
LetA=diag(A
1,...,Ak),B=(B
T
1
,...,B
T
k
)
T
,whereA iandB iaren i×ni
andn i×mmatrices overF(z) respectively,i=1,...,k, and det (λI−A i)and
det (λI−A
j) are relatively prime,iﬃ=j.Then(A,B) is controllable overF(z)
if and only if (A
i,Bi) is controllable overF(z). Corresponding conclusion over
the real ﬁeld is contained in it. For example, letA=diag(λ
1,...,λn)and
B=(b
T
1
,...,b
T
n
)
T
are respectivelyn×nandn×mmatrices over the real
ﬁeld (special RFM). If the eigenvalues ofAare diﬀerent with each other,
that is,λ
iﬃ=λj,iﬃ=j,1ﬃi, jﬃn, it implies thatλ−λ iandλ−λ jare
relatively prime. By the above conclusion we have that (A, B) is controllable
if and only if (λ
i,bi) is controllable, that is,b iﬃ=0,i=1,...,n,whichisa
well-known criterion over the real ﬁeld.
The properties overF(z) are only related to system structure, and the
systems with nonlinear elements maybe have the same or similar structure.

8 1 Introduction
For example, if a resistor in a linear network being structural controllable
is replaced by a nonlinear resistor, thenthe incidence matrix describing the
network structure is not changed. Is this nonlinear network structural con-
trollable? In this book, we propose the quasi-structural controllability and
observability conception to analyze controllability and observability of non-
linear systems and networks (see chapter 5), which makes linearity and non-
linearity closely related.
References
Anderson B D O, Hong H M (1982) Structural Controllability and Matrix nets.
International J Control, 35, 397 – 416
Balabanian N, Bickart T A (1969) Electrical Network Theory. Wiley, New York
Chen C T (1984) Linear System Theory and Design. Holt, Rinehart and Winston,
New York
Chen D B (1987) Introduction to Network Analysis. Ports and Telecom Press (in
Chinese)
Chen W K (1976) Applied Graph Theory. North-Holland, Amsterdam
Chen W K, Wu X Y, Wu S M (1992) Modern Network Analysis. Ports and Telecom
Press, Beijing (in Chinese)
Corpmat J P, Morse A S (1976) Structurally Controllable and Structurally Canon-
ical Systems IEEE Trans, AC – 21, 129 – 131
Davison E J (1977) Connectability and Structural Controllability of Composite
Systems. Automatica, 13, 109 – 123
Glover K, Silverman L M (1976) Characterization of Structural Controllability.
IEEE Trans, AC – 21, 534 – 537
Hosoe S, Matsumoto K (1979) On the Irreducibility Condition in the Structural
Controllability Theorem. IEEE Trans, AC – 24, 963 – 966
Kalman R E, Falb P L, Arbib M A (1969) Topics in Mathematical System Theory.
McGraw-Hill, New York
Lin C T (1974) Structural Controllability. IEEE Trans, AC – 19, 201 – 208
Li K, Xi Y G (1996) Oriented Graph Based Method for Testing Structural Con-
trollability. Control Theory and Applications, 13, 802 – 806 (in Chinese)
Lu K S, Wei J N (1991) Rational Function Matrices and Structural Controllability
and Observability. IET Control Theory and Applications, 138, 388 – 394
Lu K S, Wei J N (1994) Reducibility Condition of a Class of Rational Function
Matrices. SIAM J Matrix Anal Appl, 15, 151 – 161
Lu K S, Lu K (1998) Separability and Reducibility Criteria of RLC Networks over
F
zand Their Applications. International J Circuit Theory Appl, 26, 65 – 79
Lu K S (2003) Some structural Conditions under which an RLC Network is Con-
trollable overF(z). Proc IEEE ISCAS (International Symposium on Circuits
and Systems), I1, 21 – 24
Lu K S, Gao G Z (2005) The Node Voltage Equations and Structural Conditions of

References 9
Observability for RLC Networks over F(z). Proc IEEE ISCAS (International
Symposium on Circuits and Systems), 1, 764 – 767
Lu K S (2001) The Controllability of Linear Systems overF(z). Proc 40th IEEE
CDC (Conference on Decision and Control), 2676 – 2681
Mayeda H (1981) On Structural Controllability Theorem. IEEE Trans, AC – 26,
795 – 798
Murota K (1993) Mixed MatricesΔIrreducibility and Decomposition. Springer, New
York.
Murota K (1989a) On the Smith Normal Form of Structured Polynomial Matrices.
SIAM J Matrix Anal Appl, 12, 747 – 765
Murota K (1987) System Analysis by Graphs and Matroids – Structural Solvability
and Controllability. Springer, New York
Murota K (1998) On the Degree of Mixed Polynomial Matrices. SIAM J Matrix
Anal Appl, 20, 196 – 227
Murota K (1989b) On the Irreducibility of Layered Mixed Matrices. Linear and
Multilinear Algebra, 24, 273 – 288
Shields R W, Pearson J B (1976)Structural Controllability of Multiinput Linear
systems. IEEE Trans, AC – 21, 203 – 212
Wonham W M (1979) Linear Multivariable Control: A Geometric Approach.
Springer-Verlag, New York
Willems J L (1986) Structural Controllability and Observability. Syst Contr Lett,
5–12
Yamada T, Luenberger D G (1985) Generic Properties of Column-structured Ma-
trices. Linear Algebra Appl, 65, 186 – 206
Zheng D Z (2002) Linear System Theory (Second Edition). Tsinghua University
Press, Beijing (in Chinese)

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2 Matrices over FieldF(z)ofRational
Functions in Multi-parameters
In this chapter, we extend matrix theory over the real ﬁeld (main references
are [Xie, 1978] and [Gilbert, 2003]) toF(z); discuss the reducibility condition
of a class of square matrix overF(z) and its characteristic polynomial; deﬁne
type-1 matrix and two basic properties, and prove that type-1 matrix has
these two properties; give introductionto variable replacement condition for
independent parameters.
2.1 Polynomials over FieldF(z)orRingF(z)[λ]
Deﬁnition 2.1Polynomial inλ, whose coeﬃcients are members inF(z),
is called the polynomial overF(z)orringF(z)[λ]. Such as
3z
1+z
2
2
z3
z4z5+z3z
2
4
λ
2
+5λ+(z 1+z2)∈F(z)[λ].
LetFdenote any ﬁeld (may be the ﬁeld of numbers orF(z)).
Deﬁnition 2.2Polynomialf(λ),whosedegreeisnolessthan1,overthe
ﬁeldF, is called an irreducible polynomial over the ﬁeldFif it cannot be
denoted by the product of two polynomials inλ, whose degrees are less than
that off(λ); otherwise, it is reducible.
According to this deﬁnition, a one-degree polynomial is always irreducible.
The polynomial, whose degree is more than 1, may have diﬀerent results over diﬀerent ﬁelds. For example, a polynomialλ
2
+1 is an irreducible polynomial
over the ﬁeld of rational numbers, that is, it cannot be decomposed to two
one-degree polynomials inλ, whose coeﬃcients are rational numbers; and the
same as the real ﬁeld; but it is reducible over the ﬁeld of complex numbers:
λ
2
+1=(λ+j)(λ−j).
For then-degree (n>1) polynomial over the ﬁeld of numbers, it is always
denoted by the product ofnone-degree polynomials over the ﬁeld of complex
numbers. This is a well known conclusion. The degree of irreducible polyno-

12 2 Matrices over FieldF(z) of Rational Functions in Multi-parameters
mials over the ﬁeld of complex number is 1, and the degree of irreducible
polynomials over the real ﬁeld is 1 or 2.
It is a complicated problem whether ann-degree polynomial inλover
F(z) is reducible. Paper [Lu et al., 1994] explored an irreducible condition
of characteristic polynomials of a class of matrices overF(z). This condition
shows that inF(z)[λ] there exist irreducible polynomials, whose degree can
be arbitrary large (refer to Section 2.6).
2.2 Operations and Determinant of Matrix overF(z)
Matrix operation over any ﬁeld (includingF(z)) is the same as that over the
ﬁeld of numbers. The determinant calculation of a square matrix over any
ﬁeld (includingF(z)) is the same as that over the ﬁeld of numbers.
Let matrixM=(m
ij),i, j=1,2,...,n. Then the determinant deﬁnition
ofMis
detM=
δ
(−1)
τ(j1j2...jn)
m1j1m2j2...mnjn
where detMor|M|denotes the determinant ofM,j 1j2...jnis a permuta-
tion of 1,2,...,n,τ(j
1j2...jn) denotes the inverted sequence of permutation
j
1j2...jn. Such as
det

a
11a12
a21a22

=a
11a22−a12a21.
2.3 Elementary Operations of Matrices
overF(z) and Some Conclusions
Deﬁnition 2.3The change of following 3 matrices is called elementary op-
eration of matrix:
1) multiply row (column)iof matrix by any nonzero memberαλ=0of
F(z);
2) addμtimes row (column)iof matrix to row (column)jof the same
matrix whereμ∈F(z);
3) interchange rows (columns)iandjof matrix.
Corresponding to Deﬁnition 2.3, we have the following deﬁnition.
Deﬁnition 2.4The following 3 square matrices are called elementary ma-
trices:

2.3 Elementary Operations of Matrices overF(z) and Some Conclusions 13
1) multiply row (column)iof identity matrixIby any nonzero member
αﬃ=0ofF(z). The obtained matrix is called the ﬁrst elementary matrix:
⎛
⎜
⎜
⎜
⎜
⎜
⎝
10
.
.
.
α
.
.
.
01
⎞
⎟
⎟
⎟
⎟
⎟
⎠
←thei-row
↑
thei-column
2) addμtimes row (column)iof identity matrixIto row (column)jof
the same identity matrix whereμ∈F(z). The obtained matrix is called the
second elementary matrix:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
10
.
.
.
1
.
.
.
.
.
.
μ ...1
.
.
.
01
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
←thei-row
←thej-row
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
10
.
.
.
1...
μ
.
.
.
.
.
.
1
.
.
.
01
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
↑↑
theith thejth
column column
3) interchange rows (columns)iandjof identity matrixI.Theobtained
matrix is called the third elementary matrix:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
10
.
.
.
0...1
.
.
.
.
.
.
.
.
.
1...0
.
.
.
01
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
←thei-row
←thej-row
.
↑↑
theith thejth
column column

14 2 Matrices over FieldF(z) of Rational Functions in Multi-parameters
Remark 2.1For the elementary operationsof matrices over the ﬁeld of
numbers,αﬃ=0andμare arbitrary constants; however, for the elementary
operations of matrices overF(z),αﬃ=0andμare extended to arbitrary
members ofF(z) (including real constant). Similarly to elementary operations
of matrices over the ﬁeld of numbers, the following conclusions are obvious.
Theorem 2.1SupposeAis any matrix overF(z). Performing an elemen-
tary row operation toAis equal to left-multiplyingAby the same type of
elementary matrix; performing an elementary column operation toAis equal
to right-multiplyingAby the same type of elementary matrix.
Theorem 2.2Any elementary matrix has an inverse matrix, which is also
the same type elementary matrix.
Theorem 2.3SupposeAis arbitrary matrix. If we deriveBfromAby
performing some elementary row operations toA, then there must exist a
nonsingular matrixPsuch thatPA=B;ifwederiveCfromAby per-
forming some elementary column operations toA, then there must exist a
nonsingular matrixQsuch thatAQ=C;ifwederiveDfromAby perform-
ing some elementary row and column operations toA, then there must exist
nonsingular matricesPandQsuch thatPAQ=D.
Theorem 2.4Nonsingular matrix is equal to the product of some elemen-
tary matrices.
Theorem 2.5For any matrixAoverF(z),
A=rankA
T
A=rankAA
T
always hold, that is, the ranks ofA,A
T
AandAA
T
are equal, where rankA
denotes the rank ofA,A
T
denotes the transpose matrix ofA.
2.4 Operation and Canonical
Form of Matrix overF(z)
The operation and canonical form of matrix overF(z) will be presented in
this section.
2.4.1 Matrix overF(z) and its canonical expression
Amatrix(f
ij(λ))m×nwhose entries are polynomials inλover the ringF(z)[λ]
is called the matrix overF(z)[λ], shortened byA(λ). Because the member
(including constant) overF(z) can be regarded as zero-degree polynomial in
λ, the matrices overF(z)[λ] are the extension of that overF(z). For the sum,
diﬀerence and product of polynomials inλare also polynomials inλ,wecan

2.4 Operation and Canonical Form of Matrix overF(z)15
give deﬁnitions of operations for matrices overF(z)[λ]; we can also have the
determinant, minor, cofactor of square matrix overF(z)[λ], and have sub-
block, minor of general matrix overF(z)[λ]. Generally, the determinant of
square matrix overF(z)[λ]isapolynomialf(λ), but sometimes it may be a
member overF(z). Such as
σ
σ
σ
σ
σ
σ
σ
λλ +
z
1
z2
λ+
z
1
z2
λ
σ
σ
σ
σ
σ
σ
σ
=−λ
z
1
z2
−
z
2
1
z
2
2
,
σ σ σ σ
σ
λλ +z
1
λ−z 1 λ
σ
σ
σ
σ
σ
=z
2
1
,
σ
σ
σ
σ
σ
λλ
2
00
σ
σ
σ
σ
σ
=0.
So the conception of rank is intuitive.The ﬁrst and second matrices above
have full ranks, but the third does not have full rank.
We can use elementary operations to matrix overF(z) to reduce it. But
for the 2nd elementary operations, theμinF(z) in Section 2.3 should be
replaced by polynomialμ(λ)inF(z)[λ], and the 1st and 3rd elementary op-
erations are the same as Section 2.3. Thedeterminants of elementary matrices
corresponding to elementary operations are that the 1st is alsoαﬃ=0,and
the 2nd is also 1, the 3rd is also−1. In one word, they are nonzero members
overF(z). So we have the following deﬁnition.
Deﬁnition 2.5Thesquarematrixover F(z)[λ], whose determinant is
nonzero member overF(z), is called elementary matrix overF(z)[λ], or uni-
modular matrix overF(z)[λ].
The nonsingular matrix overF(z) in Section 2.3 is special elementary
matrix overF(z)[λ]. In addition, the product and direct sum (P(λ)˙+Q(λ))
of elementary matrices overF(z)[λ] are also elementary matrices overF(z)[λ].
T
he square matrixA(λ)overF(z)[λ] has an adjoint matrix
ﬃ
A(λ), and it
is also a matrix overF(z)[λ]. Especially whenA(λ) is an elementary matrix
overF(z)[λ],|A(λ)|=αﬃ=0,α∈F(z), we know
1
α
ﬃ
A(λ)isalsoamatrixover
F(z)[λ]andA(λ)
1
α
ﬃ
A(λ)=
1
α
ﬃ
A(λ)A(λ)=I. That is, elementary matrix over
F(z)[λ] always has inverse matrix and the inverse matrix is also an elementary
matrix overF(z)[λ]. For a general square matrix overF(z)[λ], we need not
to consider whether it has inverse matrix or not, even if its determinant is
equal to a nonzero polynomialf(λ), except thatf(λ)=αﬃ=0,α∈F(z),
that is, it is an elementary matrix overF(z)[λ].
Deﬁnition 2.6Suppose thatA(λ)andB(λ)aretwom×nmatrices over
F(z)[λ]. If there exist elementary matricesP(λ)andQ(λ) such that
P(λ)A(λ)Q(λ)=B(λ), we say thatA(λ)i
B(λ), and it is
denoted byA(λ)
∼
=B(λ).

16 2 Matrices over FieldF(z) of Rational Functions in Multi-parameters
Proposition 2.1The equivalent relationships of matrix overF(z)[λ]have
reﬂexivity, symmetry, and transitivity.
Because the identity matricesI
mandI nare elementary matrices over
F(z)[λ], we know thatA(λ)
∼
=A(λ)byI
mA(λ)I n=A(λ). IfA(λ)
∼
=B(λ),
P(λ)A(λ)Q(λ)= B(λ), then we know that B(λ)
∼
=A(λ)from
P(λ)
−1
B(λ)Q(λ)
−1
=A(λ). IfA(λ)
∼
=B(λ),B(λ)
∼
=C(λ), then we know
thatP
2(λ)P 1(λ)A(λ)Q 1(λ)Q 2(λ)=C(λ)fromP 1(λ)A(λ)Q 1(λ)=B(λ)and
P
2(λ)B(λ)Q 2(λ)=C(λ); and the product of elementary matrices overF(z)[λ]
is also elementary matrix overF(z)[λ], soA(λ)
∼
=C(λ).
Proposition 2.2IfA
1(λ)
∼
=B 1(λ),A 2(λ)
∼
=B 2(λ), then the direct sum
(A
1(λ)˙+A 2(λ))
∼
=(B 1(λ)˙+B 2(λ)).
BecauseP
1(λ)A 1(λ)Q 1(λ)=B 1(λ)andP 2(λ)A 2(λ)Q 2(λ)=B 2(λ), we
get that (P
1(λ)˙+P 2(λ))(A 1(λ)˙+A 2(λ))(Q 1(λ)˙+Q 2(λ)) = (B 1(λ)˙+B 2(λ)).
The direct sum of elementary matrices overF(z)[λ] is also elementary ma-
trices overF(z)[λ], so the proposition holds.
Moreover, from Proposition 2.1 we have the following proposition.
Proposition 2.3ForamatrixA(λ)overF(z)[λ], whatever elementary op-
eration to be chosen, either row or column, or how many times to be taken,
the matrix derived overF(z)[λ] is identically equivalent toA(λ).
As for rank, we also have the following proposition.
Proposition 2.4Rank(A(λ)B(λ))ﬃmin(rank(A(λ)), rank(B(λ))), this
means the rank of (A(λ)B(λ)) is no larger than the minimum rank ofA(λ)
andB(λ). Then we know that the rank does not change if we multiply a
matrix overF(z)[λ] by an elementary matrix overF(z)[λ], and the equivalent
matrices overF(z)[λ] have the same rank.
IfP(λ)
F(z)[λ]andP(λ)A(λ)=B(λ),
then we know rank(B(λ))ﬃrank(A(λ)), and byA(λ)=P(λ)
−1
B(λ)we
know rank(A(λ))ﬃrank(B(λ)). So rank(A(λ))=rank(B(λ)). Similarly, when
takingP(λ) right-multiplyA(λ), the rank is not changed.
Theorem 2.6AmatrixA(λ)overF(z)[λ], whose rank isr, can be trans-
formed by elementary operations some times to a diagonal matrix overF(z)[λ]
with the form of
D(λ)=
⎛
⎜
⎜
⎜
⎜
⎝
ϕ
1(λ)0
.
.
.
ϕ
r(λ)
00
⎞
⎟
⎟
⎟
⎟
⎠
,
whereϕ
1(λ)|ϕ 2(λ)|...|ϕ r(λ), the leading coeﬃcient of eachϕ i(λ)is1.

2.4 Operation and Canonical Form of Matrix overF(z)17
Here the notation “|” denotes the meaning of aliquot.A(λ)andD(λ)may
not be square matrices.
ProofFor rank, use induction method. When the rank ofA(λ)is0,itisa
zero matrix, which is a diagonal matrix. For the matrix with the rankr−1,
suppose that Theorem 2.6 holds. Now let us see the matrixA(λ)withrank
r.
Now considering that after any kind and any time elementary operations
forA(λ), we get all the matrices overF(z)[λ]. Among these nonzero entries
(nonzero polynomials) in these matrices, we can ﬁnd a minimum degree entry
and suppose that it isϕ
1(λ). For the matrix overF(z)[λ] includingϕ 1(λ), we
use the 3rd elementary operation to moveϕ
1(λ)tothetopleftcorner,and
A(λ) can be transformed by elementary operations some times to the form
of

ϕ
1(λ)f(λ)...
............

. (2.1)
If the leading coeﬃcient (that is, the highest degree term) ofϕ
1(λ)isnot
1 butα, we multiply the 1st row by 1/α, this is equal to make the leading
coeﬃcient to be 1 by using the ﬁrst elementary operation one time. So without
loss generality, suppose that the leading coeﬃcient is already 1. Now we can
aﬃrm that
ϕ
1(λ)|f(λ).
If not so, we useϕ
1(λ) dividingf(λ), and get the quotientq(λ)andthe
remainderr(λ) which is not zero and whose degree is less thanϕ
1(λ), and
f(λ)=q(λ)ϕ
1(λ)+r(λ).
If add−q(λ) times of the ﬁrst column to the 2nd column in Eq. (2.1) (which
is the second elementary operation), we transformA(λ)tosuchamatrix:
the entry in the 1st row and 2nd column isr(λ), but the degree ofr(λ)isless
thanϕ
1(λ), which is a contradiction. Soϕ 1(λ)|f(λ) hold. Similarlyϕ 1(λ)can
be aliquot to all the entries in the 1st row and 1st column. Taking several
times of the 2nd elementary operation, thenA(λ) can be with the form of

ϕ
1(λ)0
0A
1(λ)

. (2.2)
Because the rank of matrix (2.2) is stillr,rankA
1(λ)=r−1. By the
induction, we know that after several elementary operations,A
1(λ)canbe

18 2 Matrices over FieldF(z) of Rational Functions in Multi-parameters
with the form of
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
ϕ
2(λ)0
.
.
.
ϕ
r(λ)
00
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
, (2.3)
whereϕ
2(λ)|ϕ 3(λ)|...|ϕ r(λ) and all the leading coeﬃcients are 1. So after
several elementary operationsA(λ) can be the form of
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
ϕ
1(λ)0
ϕ
2(λ)
.
.
.
ϕ
r(λ)
00
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (2.4)
Finally we should show thatϕ
1(λ)|ϕ 2(λ), which is obvious. We can add 1
times of the 2nd row of (2.4) to 1st row, and then use the ascertained fact
above to knowϕ
1(λ)|ϕ 2(λ). The induction method is completed. Δ
Conclusion 2.1The identity matrixIcan be derived from any elementary
matrix overF(z)[λ] by performing some elementary operations.
Because the rank ofn-order elementary matrixP(λ)overF(z)[λ]isn,by
Theorem 2.6 we know that the obtained matrix by elementary operations on
P(λ)iswiththeformof
⎛
⎜
⎜
⎝
ϕ
1(λ)0
.
.
.
0 ϕ
n(λ)
⎞
⎟
⎟
⎠
and each leading coeﬃcient ofϕ
i(λ)is1.
When taking the three elementary operations onP(λ), the determinants
of matrices derived areα|P(λ)|,|P(λ)|,−|P(λ)|respectively. Because|P(λ)|
is a nonzero member overF(z), the determinant of matrix derived fromP(λ)
by several elementary operations is still a member overF(z)andwithoutλ.
Since the leading coeﬃcient ofϕ
i(λ)is1,ϕ 1(λ)ϕ 2(λ)...ϕ n(λ)=1holds,
that is,ϕ
1(λ)=...=ϕ n(λ) = 1; otherwiseϕ 1(λ)ϕ 2(λ)...ϕ n(λ) is a nonzero
degree polynomial inλ, which is a contradiction. So Conclusion 2.1 holds.
Performing an elementaryoperation on a matrix overF(z)[λ]isequalto
left-multiplying or right-multiplying the matrix overF(z)[λ]byanelementary

2.4 Operation and Canonical Form of Matrix overF(z)19
matrix. This fact is used when proving Proposition 2.3. The only diﬀerence
is that in the 2nd elementary matrix there is a polynomialμ(λ). The 1st and
3rd elementary matrices are the same as before (Section 2.3). By Conclusion
2.1 and Theorem 2.4 in Section 2.3, we have the following conclusions.
Conclusion 2.2The elementary matrix overF(z)[λ] is equal to product
of several elementary matrices (special matrices overF(z)[λ]).
Conclusion 2.3IfA(λ)
∼
=B(λ), thenB(λ) can be derived fromA(λ)by
several elementary operations.
In the following we discuss the relationship betweenϕ
i(λ)andA(λ)in
Theorem 2.6, and ﬁrst we have the following deﬁnition.
Deﬁnition 2.7Suppose that the rank ofA(λ)isr.If1ﬃsﬃr,thereexist
nonzeros-order minors inA(λ). LetD
s(λ) be the highest degree common
factor of all nonzeros-order minors inA(λ). Its leading coeﬃcient is 1. Then
there arerpolynomialsD
1(λ),D 2(λ),...,D r(λ), which are determined by
A(λ) and called the determinant factors ofA(λ).
Proposition 2.5The determinant factors of matrixA(λ)overF(z)[λ],
whose rank isr, has the propertyD
1(λ)|D 2(λ)|...|D r(λ).
BecauseD
s−1(λ) can be aliquot to all the (s−1)-order minors, we can
expand thes-order minor by row or column, and know thatD
s−1(λ)canbe
aliquot to all thes-order minors. So we have
D
s−1(λ)|D s(λ).
According to Proposition 2.5, we know
D
1(λ),
D
2(λ)
D1(λ)
,
D
3(λ)
D2(λ)
,...,
D
r(λ)
Dr−1(λ)
(2.5)
are polynomials and the leading coeﬃcients are all 1. So therpolynomials
are determined byA(λ). We have the following deﬁnition.
Deﬁnition 2.8Theserpolynomials in Expression (2.5) are called invariant
factors ofA(λ)whereris the rank ofA(λ).
Following we will prove thatϕ
1(λ),ϕ 2(λ),...,ϕr(λ) are just the invariant
factors ofA(λ) in Theorem 2.6. First we have the following proposition:
Proposition 2.6Suppose thatC(λ)=A(λ)B(λ).Accordingtotherank
we know that ifC(λ)hastdeterminant factorsD
1(λ)C,...,Dt(λ)C(that is,
rankC(λ)=t), thenA(λ)andB(λ)haveatleasttdeterminant factors
D
1(λ)A,...,Dt(λ)AandD 1(λ)B,...,Dt(λ)Brespectively.D s(λ)A|Ds(λ)C
andD s(λ)B|Ds(λ)C,s=1,...,thold.

20 2 Matrices over FieldF(z) of Rational Functions in Multi-parameters
According to the rule of the matrix multiplication, we can easily know
that eachs-order minor ofC(λ)isequalto
δ
a(λ)b(λ), wherea(λ)andb(λ)
ares-order minors ofA(λ)andB(λ) respectively. SoD
s(λ)AandD s(λ)Bare
both aliquot to eachs-order minor ofC(λ), and of course toD
s(λ)C.
Proposition 2.7Equivalent matrices overF(z)[λ] have not only the same
rank, but also the same determinant factors and invariant factors.
The only fact we should point out is thatB(λ) derived by using an el-
ementary matrixP(λ)overF(z)[λ] left-multiplying or right-multiplying a
matrixA(λ)overF(z)[λ] has the same determinant factors asA(λ).
Suppose thatB(λ)= P(λ)A(λ). By Proposition 2.6 we have
D
s(λ)A|Ds(λ)B.ByA(λ)=P(λ)
−1
B(λ)weknowD s(λ)B|Ds(λ)A. Because
all the leading coeﬃcients are 1,D
s(λ)A=Ds(λ)Bmust hold. It is the same
result when right-multiplying.
Theorem 2.7ϕ
1(λ),...,ϕr(λ) in Theorem 2.6 are justrinvariant factors
ofA(λ).
ProofBy Proposition 2.7 we know that the invariant factors ofA(λ)isthe
same as that of the equivalent diagonal matrixD(λ)overF(z)[λ]. According
toD(λ) in Theorem 2.6 we easily know (note thatϕ
1(λ)|...|ϕ r(λ)) that its
determinant factors are
D
1(λ)=ϕ 1(λ),D 2(λ)=ϕ 1(λ)ϕ 2(λ),...,D r(λ)=ϕ 1(λ)...ϕ r(λ),
Immediately we can get the invariant factorsϕ
1(λ),ϕ 2(λ),...,ϕr(λ).ξ
According to the above, the diagonal matrix overF(z)[λ] in Theorem 2.6
determined byA(λ) is called typical form or canonical expression.
Following we decomposeA(λ) to its invariant factorsϕ
1(λ),...,ϕr(λ)
overF(z). Suppose that
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
ϕ
1(λ)=[e 1(λ)]
n11
[e2(λ)]
n21
...[e k(λ)]
nk1
ϕ2(λ)=[e 1(λ)]
n12
[e2(λ)]
n22
...[e k(λ)]
nk2
............
ϕ
r(λ)=[e 1(λ)]
n1r
[e2(λ)]
n2r
...[e k(λ)]
nkr
, (2.6)
wheree
1(λ),e2(λ),...,ek(λ) are diﬀerent nondecomposed polynomials over
F(z) with their leading coeﬃcients of 1; exponentialn
ij(i=1,...,k;j=
1,...,r) is nonnegative integer (of course there may be some zeros, but each
of exponentialsn
1r,...,nkrin the last row is not zero).
According to the “in turn aliquot” of invariant factors, we know
n
11λn12λ...λn 1r;...;n k1λnk2λ...λn kr. (2.7)

2.4 Operation and Canonical Form of Matrix overF(z)21
Deﬁnition 2.9These factors [e i(λ)]
nij
above, except that exponentials are
zeros, are called elementary divisors ofA(λ). When counting the number of
elementary divisors, the repeated onesshould be included. All the elementary
divisors are called elementary divisor group ofA(λ).
For example, over the ﬁeld of rational numbers (the sub-ﬁeld ofF(z)),
supposer=2,
⎧
⎨
⎩
ϕ
1(λ)=(λ−1)
2
(λ
2
+λ+3)
3
ϕ2(λ)=(λ−1)
2
(λ
2
+λ+3)
5
.
A(λ) has 4 elementary divisors: (λ−1)
2
,(λ−1)
2
,(λ
2
+λ+3)
3
,(λ
2
+λ+3)
5
,
which compose the elementary divisor group ofA(λ).
Clearly, the elementary divisor group ofA(λ) is determined byA(λ). After
given elementary divisor group ofA(λ), by the rule of Exponential (2.7) and
the rank ofA(λ) we can easily get the invariant factors ofA(λ).
To summarize the above contents,wehavethefollowingtheorem.
Theorem 2.8Twom×nmatricesA(λ)
∼
=B(λ)overF(z)[λ]⇔they have
the same canonical expression⇔they have the same determinant factors
⇔they have the same invariant factors⇔they have the same rank and
elementary divisor group (see Remark 2.2)⇔one can be derived from the
other by several elementary operations, where⇔denotes equivalence.
Remark 2.2Two matrices overF(z)[λ] which are not equivalent may have
the same elementary divisor group. Such that
A(λ)=

λ+1 0
0λ−1

,B(λ)=

λ
2
−10
00

have the same elementary divisor group: (λ+1),(λ−1). Clearly they are
not equivalent, because rankA(λ) = 2, but rankB(λ)=1.
From the above contents, if we want to get the elementary divisor group of
A(λ) in Theorem 2.6 (we will see the importance later), it seems that we ﬁrst
makeA(λ) to canonical expression, then get the elementary divisors, which
isnoteasy.Infact,ﬁrstwecanmakeA(λ) to diagonal form by elementary
operations (it is easier than ﬁnding canonical expression), then decompose it
to factors. In this way we can get the elementary divisors. Now we need to
prove the following Theorem.

22 2 Matrices over FieldF(z) of Rational Functions in Multi-parameters
Theorem 2.9IfA(λ) is equivalent to a diagonal matrix overF(z)[λ]
B(λ)=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
g
1(λ)0
.
.
.
g
r(λ)
00
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
, (2.8)
where all the leading coeﬃcients ofg
i(λ) are 1, which may not be the invariant
factors ofA(λ), and
g
i(λ)=[f 1(λ)]
mi
[f2(λ)]
li
...[f t(λ)]
ni
,(i=1,...,r),
wheref
1(λ),...,ft(λ) are diﬀerent nondecomposed polynomials with the
leading coeﬃcient of 1. All [f
1(λ)]
mi
,...,[f t(λ)]
ni
(i=1,...,r)composeto
elementary divisor group ofA(λ), except that exponentials are zeros.
ProofFirst let us see the powers off
1(λ). Because the equivalence is not
changed when adjusting the order ofg
i(λ) on the main diagonal, supposing
that after adjustment we havem
1ﬃm 2ﬃ...ﬃm r.So[f 1(λ)]
m1+...+m s
(sﬃ
r) is aliquot to anys-order minor ofB(λ) and also to determinant factor
D
s(λ). But [f 1(λ)]
m1+...+m s+1
is not aliquot to thes-order minor at the top
left corner neither toD
s(λ). So we have
D
s(λ)=[f 1(λ)]
m1+...+m s−1+ms
h(λ),f 1(λ) is not aliquot toh(λ).
Similarly, we have
D
s−1(λ)=[f 1(λ)]
m1+...+m s−1
k(λ),f 1(λ) is not aliquot tok(λ).
So the decomposition of invariant factorsϕ
s(λ)justcontainsthepower
[f
1(λ)]
ms
.Whens=1,...,r, it is the same result. This shows not only
that [f
1(λ)]
m1
,...,[f 1(λ)]
mr
are the elementary divisors ofA(λ) except that
the exponentials are zeros, but also that for all the powers off
1(λ),A(λ)has
such elementary divisors only.
Similarly, for all the power off
2(λ),...,ft(λ),A(λ) just have the decom-
posed elementary divisors by the theorem, except that the exponentials are
zeros.
Finally, we should point out that any polynomialf(λ)withanypower,
which is diﬀerent fromf
j(λ), cannot be the elementary divisors ofA(λ), so the
theorem is proved. It is obvious. Becausef(λ) is not aliquot tog
1(λ)g2(λ)...
g
r(λ)=D r(λ), then it is not aliquot to any invariant factors.∗

2.4 Operation and Canonical Form of Matrix overF(z)23
2.4.2 Characteristic matrix
In order to investigate the canonical form ofA,wewillusematrixtheory
overF(z)[λ], which has a close relationship to the following special matrix
overF(z)[λ]. So let us discuss this special matrix overF(z)[λ].
Deﬁnition 2.10Suppose thatA=(a
ij)isanyn-order matrix overF(z).
Then the following special matrix overF(z)[λ]withtheform
λI−A=
⎛
⎜
⎜
⎜
⎝
λ−a
11...−a 1n
.
.
.
.
.
.
.
.
.
−a
n1... λ−a nn
⎞
⎟
⎟
⎟
⎠
(2.9)
is called the characteristic matrix ofA.
After expending the determinant|λI−A|of characteristic matrix (λI−
A)ofA, we can get the characteristic polynomialf(λ)ofA.Firstweuse
characteristic matrix to prove the following theorem.
Theorem 2.10[Cayley-Hamilton Theorem] Ann-order matrixAsatisﬁes
its characteristic polynomialf(λ), that is,f(A)=0.
ProofLet
f(λ)=λ
n
−a1λ
n−1
+a2λ
n−2
−...+(−1)
n
an.
Because each entry of adjoint
Δ
(λI−A)of(λI−A) is a polynomial inλwhose
degree is no more thann−1, it can be denoted by
Δ
(λI−A)=C
1λ
n−1
+C2λ
n−2
+...+C n−1λ+C n,
whereC
1,...,Cnaren-order matrices overF(z). Then
(λI−A)(C
1λ
n−1
+...+C n)=f(λ)I=Iλ
n
−a1Iλ
n−1
+...+(−1)
n
anI.
Comparing the coeﬃcient matrices ofλwhose powers are same, we can get
C
1=I,C2−AC 1=−a 1I, C3−AC 2=a2I,...,−AC n=(−1)
n
anI.
(2.10)
UseA
n
,A
n−1
,...,A,Ileft-multiplying Eq. (2.10) respectively to get new
equations and add them together. Then the left side is
A
n
C1+A
n−1
(C2−AC1)+A
n−2
(C3−AC2)+...+A(C n−ACn−1)−AC n=0
and the right side is
A
n
−a1A
n−1
+a2A
n−2
−...+(−1)
n
anI=f(A).
Theorem is proved. Δ

24 2 Matrices over FieldF(z) of Rational Functions in Multi-parameters
Following we will prove an important theorem for researching the canon-
ical form. Now ﬁrst we consider the matrix division overF(z)[λ].
Proposition 2.8Similar as above, suppose the two square matrices over
F(z)[λ] as follows:
B(λ)=B
0λ
m
+B1λ
m−1
+...+B m,B0ﬃ=0;C(λ)=C 0λ
m
ﬃ
+C1λ
m
ﬃ
−1
+...+C m
ﬃ,C0is nonsingular. We say degreeB(λ)=m(or degB(λ)=m);
degC(λ)=m
Δ
.
Then there exists uniqueQ
1(λ),R 1(λ) such that
B(λ)=Q
1(λ)C(λ)+R 1(λ),
whereR
1(λ) is zero or degR 1(λ)<degC(λ). Then there exists unique
Q
2(λ),R 2(λ) such that
B(λ)=C(λ)Q
2(λ)+R 2(λ),
whereR
2(λ) is zero or degR 2(λ)<degC(λ).
It is similar to the general polynomial division with remainder. The only
diﬀerence is that when dividing, we should note the degree of factors to
multiply. So the left-division and right-division are diﬀerent, and also there
are two equations and two remainders.
By the way, the matrix overF(z)[λ] can be regarded as polynomial matrix,
whose entries are polynomials; it also can be regarded as matrix polynomial,
whose coeﬃcients are matrices.
Deﬁnition 2.11Suppose thatAandBare twon-order matrices. If there
exists a nonsingular matrixPoverF(z) such thatP
−1
AP=B,thenwesay
thatAis similar toB, denoted byA∼B;orwecansaythatBis derived
fromAby similarity transformation.
Proposition 2.9The similarity of square matrices is of reﬂexivity, symme-
try and transitivity.
SinceI
−1
AI=A, we know thatA∼A;ifA∼B,thenwe
get(P
−1
)
−1
BP
−1
=AbyP
−1
AP=Band knowB∼A;ifA∼B,B∼C,
then according toP
−1
AP=B,Q
−1
BQ=C,weget(PQ)
−1
A(PQ)=C
and knowA∼C. Besides these, already we have the following proposition.
Proposition 2.10The similar matrices have the same rank, characteristic
polynomial and eigenvalues.
Theorem 2.11A∼B⇔(λI−A)
∼
=(λI−B).
ProofIfA∼B,wehaveP
−1
AP=B,thenP
−1
(λI−A)P=λI−B,and
P
−1
andPare elementary matrices overF(z)[λ]. So (λI−A)
∼
=(λI−B).

2.4 Operation and Canonical Form of Matrix overF(z)25
On the other hand, if (λI−A)
∼
=(λI−B), there exist matricesP(λ)and
Q(λ)overF(z)[λ] such thatP(λ)(λI−A)Q(λ)=(λI−B). By Proposition
2.8 we have
P(λ)=(λI−B)P
1(λ)+R, Q(λ)=Q 1(λ)(λI−B)+S
whereRandSare zeros or matrices overF(z) (because its degree is less
than deg(λI−B)=1).Nowforshort,letA
0andB 0denote (λI −A)and
(λI−B) respectively. So we have
B
0=PA 0Q=(B 0P1+R)A 0(Q1B0+S)
=B
0P1A0Q1B0+B0P1A0S+RA 0Q1B0+RA 0S.
Move the last term to left, and replaceSandRwithQ−Q
1B0andP−B 0P1
respectively, yields
B
0−RA 0S=B 0P1A0Q1B0+B0P1A0(Q−Q 1B0)+(P−B 0P1)A0Q1B0
=B0P1A0Q1B0+B0P1A0Q−B 0P1A0Q1B0
+PA0Q1B0−B0P1A0Q1B0
=B0P1A0Q+PA 0Q1B0−B0P1A0Q1B0. (2.11)
SinceB
0=PA 0Q,A 0Q=P
−1
B0andPA 0=B0Q
−1
. Substituting the two
equations to Eq. (2.11) yields
B
0−RA 0S=B 0P1P
−1
B0+B0Q
−1
Q1B0−B0P1A0Q1B0
=B0(P1P
−1
+Q
−1
Q1−P1A0Q1)B0.
Now letG(λ) denote the matrix in bracket, and then rewrite it using the
former notation, then
(λI−B)−R(λI−A)S=(λI−B)G(λ)(λI−B).
Observe the degrees ofλin two sides of equation: the degree of left side
should be no large than 1; ifG(λ)ﬃ= 0, then the degree of right side should
be no less than 2. This is a contradiction. SoG(λ) = 0, and
λI−B=R(λI−A)S=λRS−RAS.
Then we knowRS=I,R=S
−1
andB=RAS=S
−1
AS.SoA∼B.Δ
Let us see the canonical expression ofλI−A. Because|λI−A|=f(λ)is
ann-degree polynomial, the rank ofλI−Aisn. According to Theorem 2.6

26 2 Matrices over FieldF(z) of Rational Functions in Multi-parameters
we know that there exist elementary matricesP(λ)andQ(λ)overF(z)[λ]
such that
P(λ)(λI−A)Q(λ)=
⎛
⎜
⎜
⎝
ϕ
1(λ)0
.
.
.
0 ϕ
n(λ)
⎞
⎟
⎟
⎠
,ϕ
1(λ)|...|ϕ n(λ).
If we take determinants to both sides, then according to that|P(λ)|=aﬃ=0,
|Q(λ)|=bﬃ= 0 and all the leading coeﬃcients off(λ)andϕ
i(λ)are1we
have
f(λ)=ϕ
1(λ)...ϕ n(λ),(ab=1musthold)
Because the rank off(λ)isnand there arenfactors at right side, which can
be aliquot in turn; generally several (such asn−k) invariant factors in front
are all 1 and the lastkpolynomials containλ.Thatis
(λI−A)
∼
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
10
.
.
.
1
ϕ
1(λ)
.
.
.
0 ϕ
k(λ)
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,ϕ
1(λ)|...|ϕ k(λ),
ϕ
1(λ)...ϕ k(λ)=|λI−A|.
Furthermore, we have the following theorem.
Theorem 2.12If the elementary divisor group of (λI−A)isψ
1(λ),
ψ
2(λ),...,ψt(λ), then
(λI−A)
∼
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
10
.
.
.
1
ψ
1(λ)
ψ
2(λ)
.
.
.
0 ψ
t(λ)
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
According to canonical expression above and using the following proposition
repeatedly, it can be proved.

2.4 Operation and Canonical Form of Matrix overF(z)27
Proposition 2.11Ifg(λ)andh(λ) is relatively prime, then

g(λ)0
0h(λ)

∼
=

10
0g(λ)h(λ)

.
Because there existϕ(λ)andψ(λ) such thatg(λ)ϕ(λ)+h(λ)ψ(λ)=1,
we have
σ
σ
σ
σ
σ
σ
1 −1
h(λ)ψ(λ)g(λ)ϕ(λ)
σ
σ
σ
σ
σ
σ
=
σ
σ
σ
σ
σ
σ
ϕ(λ)h(λ)
−ψ(λ)g(λ)
σ
σ
σ
σ
σ
σ
=1.
Acco
rding to the following
⎛
⎝
1 −1
h(λ)ψ(λ)g(λ)ϕ(λ)
⎞
⎠
⎛
⎝
g(λ)0
0h(λ)
⎞
⎠
⎛
⎝
ϕ(λ)h(λ)
−ψ(λ)g(λ)
⎞
⎠=
⎛
⎝
10
0g(λ)h(λ)
⎞
⎠,
we know that the proposition holds.
2.4.3 Two canonical forms of nonderogatory matrix
From this section we begin to discuss the canonical form of general square
matrix under transformation of similitude.
First we discuss the canonical form of a special class of matrix (that is,
nonderogatory matrix) and its characters.
Deﬁnition 2.12If the invariant factors of the characteristic matrix ofAare
1,...,1,ϕ(λ)=|λI−A|,t
henwecallAthe nonderogatory matrix belonging
toϕ(λ).
Theorem 2.13If am-order matrixAis a nonderogatory matrix belonging
toϕ(λ)and
ϕ(λ)=λ
m
+a1λ
m−1
+a2λ
m−2
+...+a m−1λ+a m
thenAmust be similar to the matrix determined byAas follows
(I) C=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
00...0−a
m
10...0−a m−1
.
.
.1
.
.
.
.
.
.
0
.
.
.
.
.
.0−a
2
00...1−a 1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.

28 2 Matrices over FieldF(z) of Rational Functions in Multi-parameters
Cis still a nonderogatory matrix belonging toϕ(λ), and called the canonical
nonderogatory form (I) belonging toϕ(λ).
ProofFirst we have
λI−C=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
λ00 ...0 a
m
−1λ0...0a m−1
0−1λ ...0a m−2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
000 λa
2
000 ...−1λ+a 1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
Now we consider|λI−C|and addλ, λ
2
,...,λ
m−1
timesofthe2nd,3rd,...,
mth row respectively to the 1st row, then the determinant is not changed and
|λI−C|=
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
00 ...0ϕ(λ)
−1λ ...0a
m−1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
00 ···λa
2
00 ...−1λ+a 1
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
.
So themth determinant factor of (λI−C)isϕ(λ). But the (m−1)-order
minor of (λI−C)atthebottomleftcorneris(−1)
m−1
. So we know that
the (m−1)th determinant factor is 1. The invariant factors of (λI−C)must
be 1,...,1,ϕ(λ)=|λI−C|,andCis a nonderogatory matrix belonging to
ϕ(λ). Also since (λI−A)
∼
=(λI−C), we know thatA∼C. Δ
According to Theorem 2.10 we know that for any square matrixAthere
exists at least one nonzero polynomialf(λ) such thatf(A)=0,whichresults
in a question whether there is any other polynomial with this property. Ob-
viously it is true, such asg(λ)f(λ) which is arbitrary times off(λ), because
g(A)f(A)=g(A)·0 = 0. We can get the following deﬁnition.
Deﬁnition 2.13Suppose thatAis a square matrix. If a polynomialg(λ)
with its leading coeﬃcient being 1 such thatg(A) = 0, but any nonzero
polynomialh(λ) whose degree is less than that ofg(λ)d

property, that is,h(A)ﬃ=0,theng(λ) is called the minimum polynomial
ofA.

2.4 Operation and Canonical Form of Matrix overF(z)29
Proposition 2.12Ifg(λ) is the minimum polynomial ofAandh(λ)isany
polynomial. Ifh(A)=0,theng(λ)|h(λ)holds.
In fact, takeh(λ) divided byg(λ), if the remainderr(λ)ﬃ=0,thenits
degree is less than that ofg(λ). According toh(λ)=g(λ)q(λ)+r(λ)wehave
r(λ)=h(λ)−g(λ)q(λ).
SubstituteAto it and yield
r(A)=h(A)−g(A)q(A)=0−0q(A)=
.
This is a contradiction. Sog(λ)|h(λ) must hold.
Proposition 2.13Similar matrices have the same minimum polynomial.
Suppose thatA∼Band the minimum polynomials ofAandBareg(λ)
andh(λ), respectively. Then suppose thatP
−1
AP=B. According to
g(B)=g(P
−1
AP)=P
−1
g(A)P=P
−1
0P=0,
h(A)=h(PBP
−1
)=Ph(B)P
−1
=P0P
−1
=0, (2.12)
h(λ)|g(λ)andg(λ)|h(λ) hold. As both of the leading coeﬃcients of them are
1, sog(λ)=h(λ)holds.
For easy application, we write the middle result of Eq. (2.12) as follows,
g(P
−1
AP)=P
−1
g(A)P, (2.13)
whereg(λ) is an arbitrary polynomial,Ais any square matrix.
According to Proposition 2.12 we know that the minimum polynomial of
any square matrix can be aliquot to its characteristic polynomial. Especially
we have the following theorem.
Theorem 2.14The minimum polynomial of nonderogatory matrix is the
characteristic polynomial.
ProofSuppose thatAandCare the matrices mentioned in Theorem 2.13.
By Proposition 2.13 the only thing is to prove that the minimum polynomial
ofCisϕ(λ). Of course it is only needed to prove that any nonzero polynomial
h(λ)=b
1λ
m−1
+b2λ
m−2
+...+b m−1λ+b m
(there is at least one nonzero coeﬃcient inb 1,...,bm) with its degree being
less thanmsuch thath(C)ﬃ=0.
WhenCmultiplies each columne
1,e2,...,emofIm,wehave
Ce
1=e2,C
2
e1=Ce2=e3,C
3
e1=Ce3=e4,...,
C
m−2
e1=Cem−2=em−1,C
m−1
e1=Cem−1=em

30 2 Matrices over FieldF(z) of Rational Functions in Multi-parameters
and then
h(C)e
1=b1C
m−1
e1+b2C
m−2
e1+...+b m−1Ce1+bme1
=b1em+b2em−1+...+b m−1e2+bme1
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
b
m
bm−1
.
.
.
b
2
b1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
ﬀ=0.
Soh(C)ﬀ=0holds. ﬃ
Except for the nonderogatory matrix, the degree of the minimum poly-
nomials of all othern-order matrices is less thann, which is the meaning of
nonderogatory.
According to the proof process of the above theorems, we see that it is
easy to deal the problems by the canonical form. This is the meaning why
we investigate canonical form.
Recall the form of elementary divisor, further we have the following the-
orem.
Theorem 2.15Suppose thatAis a nonderogatory matrix belonging to
ϕ(λ)andϕ(λ)=[e(λ)]
s
. Suppose thatCis the canonical nonderogatory
form (I) belonging toe(λ), and
N=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0...01
0...00
.
.
.
.
.
.
.
.
.
0...00
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
has the same order withC.SoAmust be similar to the matrix
(II) D=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
C 0
NC
N
.
.
.
.
.
.
.
.
.
0 NC
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,

2.4 Operation and Canonical Form of Matrix overF(z)31
which is determined byA.
|λI−D|=
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
λI−C 0
−NλI −C
−N
.
.
.
.
.
.
.
.
.
0 −NλI−C
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
=(e(λ))
s
=ϕ(λ).
Dis still a nonderogatory matrix belonging toϕ(λ)=[e(λ)]
s
,whichisalso
called the canonical nonderogatory form (II) belonging to
ϕ(λ)=(e(λ))
s
.
ProofSuppose that the order of bothAandDism.Observethe(m−1)-
order minor at bottom left corner, whose entries at main diagonal are all 1
(consists of the ones inCand the one inN) and left side of main diagonal
are zeros. So the (m−1)th determinant factor of (λI−D)mustbe1,and
Dis the nonderogatory matrix belonging toϕ(λ)=|λI−D|.Ofcoursewe
have (λI−A)
∼
=(λI−D), soA∼D. Δ
In order to comprehend the proof, take example as follows: Suppose that
e(λ)=λ
3
−z3λ
2
−z2λ−z 1,s=3,
so
C=
⎛
⎜
⎜
⎝
00z
1
10z 2
01z 3
⎞
⎟
⎟
⎠
,N =
⎛
⎜
⎜
⎝
001
000
000
⎞
⎟
⎟
⎠
,
D=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
00z
1
10z 2 0
01z
3
00 1 00z 1
00 0 10z 2
00 0 01z 3
00 1 00z 1
000010 z 2
00 0 01z 3
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎝
C00
NC 0
0NC
⎞
⎟
⎟
⎠
.

32 2 Matrices over FieldF(z) of Rational Functions in Multi-parameters
The main diagonal of 8-order minor at bottom left corner consists of the ones
inCand the one inN, and the entries at left side of main diagonal are zeros.
Conclusion 2.4IfAis a nonderogatory matrix belonging to (λ−a)
s
,a∈
F(z), thenAis similar to the matrix with the form of
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
a 0
1a
1
.
.
.
.
.
.
01 a
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,(s-order).
It is still a matrix belonging to (λ−a)
s
(canonical nonderogatory from (II)).
Heree(λ)=λ−a, C=(a),N= (1). The matrix is a classic Jordan
matrix.
2.4.4 Rational canonical form and general
Jordan canonical form
It is easy to get any canonical form of general square matrix based on the
contents in Section 2.4.3.
Suppose thatAis an arbitraryn-order matrix. At the same time suppose
that all the invariant factors of (λI−A)are1,...,1,ϕ
1(λ),ϕ 2(λ),...,ϕk(λ),
whereϕ
i(λ) is a polynomial with the leading coeﬃcient being 1 and
degϕ
i(λ)Δ1, 1ﬃiﬃk,andϕ 1(λ)|ϕ 2(λ)|...|ϕ k(λ); the elementary divisor
group of (λI−A)isψ
1(λ),ψ 2(λ),...,ψt(λ), each of which is a polynomial
with the form of [e(λ)]
s
,wheresΔ1ande(λ) is a nondecomposed (irre-
ducible) polynomial overF(z) with the leading coeﬃcient being 1. And we
have
ϕ
1(λ)ϕ 2(λ)...ϕ k(λ)=ψ 1(λ)ψ 2(λ)...ψ t(λ)=|λI−A|.
The degree sum of eachϕ
i(λ)isn,soisforeachψ i(λ).
Theorem 2.16Suppose that all nonconstant invariant factors of (λI−A)
areϕ
1(λ),...,ϕk(λ)andC 1,...,Ckare canonical nonderogatory form (I)
belonging toϕ
1(λ),...,ϕk(λ), respectively, thenAmust be similar to the
matrix
⎛
⎜
⎝
C
1 0
.
.
.
0 C
k
⎞
⎟
⎠,(canonical form (I))
determined byA.

2.4 Operation and Canonical Form of Matrix overF(z)33
ProofThe similarity is obtained immediately by the following equivalence.
(λI−A)
∼
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
10
.
.
.
1
ϕ
1(λ)
.
.
.
0 ϕ
k(λ)
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
∼
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
10
.
.
.
ϕ
1(λ)
.
.
.
1
.
.
.
0 ϕ
k(λ)
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
∼
=
⎛
⎜
⎜
⎜
⎜
⎝
λI−C
1 0
λI−C
2
.
.
.
0 λI−C
k
⎞
⎟
⎟
⎟
⎟
⎠
=λI−
⎛
⎜
⎜
⎜
⎜
⎝
C
1 0
C
2
.
.
.
0 C
k
⎞
⎟
⎟
⎟
⎟
⎠
.
Because all invariant factors have a certain degree and the same toC
i,the
Theorem is proved. Δ
The block diagonal matrix in this Theorem is called the canonical form
(I) ofA, or called rational canonical form. This is because the ﬁnding process
of all the invariant factors of (λI−A) consists of rational operation (add,
minus, multiply, and divide).
Similarly, we can prove the following theorem.
Theorem 2.17Suppose that the elementary divisor group of (λI−A)is
ψ
1(λ),...,ψt(λ)andD 1,...,Dtare canonical nonderogatory form (II) be-

34 2 Matrices over FieldF(z) of Rational Functions in Multi-parameters
longing toψ 1(λ),...,ψt(λ), respectively, thenAmust be similar to
⎛
⎜
⎝
D
1 0
.
.
.
0 D
t
⎞
⎟
⎠,(canonical form (II))
and this matrix is just determined byAexcept the order of eachD
i.
This matrix is called the canonical form II ofAor general Jordan canonical
form (because eachD
iis corresponding to one elementary divisorψ i,i=
1,2,...,t).
Theorem 2.18The minimum polynomial ofAis the last invariant factor
ϕ
k(λ)of(λI−A).
ProofBy canonical form (I) we have
A=P
⎛
⎜
⎝
C
1 0
.
.
.
0 C
k
⎞
⎟
⎠P
−1
.
First from Theorem 2.14 we knowϕ
1(C1)=0,...,ϕ k(Ck) = 0, and then
fromϕ
1(λ)|...|ϕ k(λ)weknowϕ k(C1)=0,ϕ k(C2)=0,...,ϕ k(Ck−1)=0.
So
ϕ
k(A)=ϕ k
⎛
⎜
⎝P
⎛
⎜
⎝
C
1 0
.
.
.
0 C
k
⎞
⎟
⎠P
−1
⎞
⎟
⎠
=P
⎛
⎜
⎝
ϕ
k(C1)0
.
.
.
0 ϕ
k(Ck)
⎞
⎟
⎠P
−1
=0.
And ifg(λ) is an arbitrary polynomial with the degree less than that of
ϕ
k(λ), then by Theorem 2.14 we knowg(C k)ﬃ= 0, and (by the rank of similar
matrices being the same)
g(A)=P
⎛
⎜
⎝
g(C
1)0
.
.
.
0 g(C
k)
⎞
⎟
⎠P
−1
ﬃ=0.
Soϕ
k(λ) is just the minimum polynomial ofA. Δ

2.5 Reducibility of Square Matrix overF(z)35
2.5 Reducibility of Square Matrix overF(z)
Deﬁnition 2.14Suppose thatAis ann×nmatrix overF(z).Ais called
a reducible matrix overF(z)ifthereexistsann×nnonsingular matrixT
overF(z) such that
TAT
−1
=

A
110
A
21A22

(the similar matrixTAT
−1
ofAis a block-triangular
matrix) whereA
11andA 22aren 1×n1andn 2×n2matrices, respectively,
n
1+n2=n,1τn 1<n;otherwiseAis irreducible overF(z).
Proposition 2.14If ann×nmatrixAoverF(z) is reducible, its charac-
teristic matrix and characteristic polynomial are both reducible.
ProofBecauseAis reducible overF(z), there exist a nonsingular matrix
Tsuch thatTAT
−1
a quasi-triangular matrix. Then we have
T(λI−A)T
−1
=λI−TAT
−1
=λI−

A
110
A
21A22

=

λI
1−A11 0
−A
21 λI2−A22

.
The characteristic matrix ofAis reducible.
Take determinant both sides of the equation above. Left side is
|T||λI−A|
σ
σT
−1
σ
σ=|λI−A|and right side is|λI
1−A11||λI2−A22|which
is the product of characteristic polynomials ofA
11andA 22, that are polyno-
mials overF(z). The characteristic polynomial ofAis a reducible polynomial
overF(z). λ
Proposition 2.15|λI−A|is irreducible polynomial inλoverF(z) if and
only ifAis an irreducible matrix overF(z).
ProofOnly suﬃciency needs to be proved. If|λI−A|is reducible, let
|λI−A|=ψ
1(λ)ψ 2(λ)...ψ k(λ),ψ i(λ)∈F(z)[λ], degψ i(λ)ϕ1, 1τiτk,
andψ
1(λ),ψ 2(λ),...,ψk(λ) be the elementary divisor group. Ifkϕ2, ac-
cording to Theorem 2.17 we know
λI−A
∼
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
10
.
.
.
1
ψ
1
.
.
.
0 ψ
k
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
∼
=
⎛
⎜
⎝
λI−A
1 0
.
.
.
0 λI−A
k
⎞
⎟
⎠

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Ranskassa

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Title: Kansalaissota Ranskassa
Author: Karl Marx
Author of introduction, etc.: Friedrich Engels
Release date: August 21, 2017 [eBook #55401]
Language: Finnish
Credits: Produced by Jari Koivisto
*** START OF THE PROJECT GUTENBERG EBOOK KANSALAISSOTA
RANSKASSA ***

Produced by Jari Koivisto
KANSALAISSOTA RANSKASSA
Kirj.
Karl Marx
Johdannon kirjoittanut Friedrich Engels
Suomennos

Helsingissä, Työväen Sanomalehti O.-Y., 1918.

SISÄLTÖ:
1. Johdanto, kirj. Friedrich Engels. 2. Kansainvälisen työväenliiton
pääneuvoston ensimäinen selostus saksalais-ranskalaisesta sodasta,
kirj. Karl Marx. 3. Saman toinen selostus siitä, kirj. Karl Marx. 4.
Saman selostus kansalaissodasta Ranskassa, kirj. Karl Marx.
Johdanto.
Kehoitus julkaista uudelleen kansainvälisen pääneuvoston julistus
"kansalaissodasta Ranskassa" ja varustaa se johdannolla tuli minulle
odottamatta. Voin siitä syystä tässä ainoastaan lyhyesti kosketella
pääkohtia.
Minä asetan pääneuvoston kaksi lyhempää selostusta saksalais-
ranskalaisesta sodasta edellämainitun pitemmän teoksen edelle.
Ensiksikin siitä syystä, että "Kansalaissodassa" viitataan toiseen
niistä, jota taas ei voi täysin ymmärtää lukematta toista. Toiseksi,
koska nämä molemmat, niinikään Marxin kirjoittamat selostukset
yhtä suuressa määrässä kuin "Kansalaissotakin" ovat erinomaisia
näytteitä kirjoittajan ihmeteltävästä, ensi kerran "Louis Bonaparten

18. Brumaire"-kirjassa koetellusta kyvystä selvästi käsittää suurten
historiallisten tapausten luonne, merkitys ja välttämättömät
seuraukset, jo aikana, jolloin nämä tapaukset vielä välähtelevät
silmäimme edessä tai juuri ovat päättyneet. Ja lopuksi, koska me
vielä tänä päivänä Saksassa saamme kärsiä noiden tapauksien
aiheuttamista, Marxin ennustamista seurauksista.
Vai eikö ole tapahtunut se, mitä sanotaan ensimäisessä
tiedonannossa, että jos Saksan puolustussota Louis Bonapartea
vastaan turmeltuu valloitussodaksi Ranskan kansaa vastaan, kaikki
se paha, mikä Saksaa kohtasi niin kutsuttujen vapautussotien
jälkeen, jälleen tulee leimahtamaan esiin uusiutuneella ankaruudella?
Eikö meillä ole ollut kaksikymmentä vuotta lisää Bismarck-hallitusta,
kansanyllyttäjien vainoamisen sijasta poikkeuslaki ja sosialistivaino,
mukana sama poliisimielivalta, kirjaimellisesti sama pääkarvoja
nostattava laintulkinta?
Ja eikö ole kirjaimelleen käynyt toteen ennustus, että Elsass-
Lothringin anastus tulisi ajamaan Ranskan Venäjän syliin? ja että
tämän anastuksen jälkeen Saksasta joko tulee Venäjän julkinen renki
tai on sen lyhyen levon jälkeen pakko varustautua uuteen sotaan,
"rotusotaan yhdistyneitä slaavilaisia ja romanilaisia rotuja vastaan?"
Eikö ole ranskalaisten maakuntien anastus ajanut Ranskan Venäjän
syliin? Eikö ole Bismarck kokonaista kaksikymmentä vuotta turhaan
kilpaillut tsaarin suosiosta tekemällä palveluksia, jotka ovat vielä
alhaisempia kuin mitä pieni Preussi oli tottunut laskemaan pyhän
Venäjän jalkain juureen, ennenkuin siitä vielä oli tullut "Europan
ensimäinen suurvalta?" Ja eikö vielä joka ikinen päivä uhkaa
Damokleen miekkana päämme päällä sota, jonka ensimäisenä
päivänä kaikki sovitut ruhtinasliitot tulevat hajoamaan kuin akanat
tuuleen, sota, jonka päättymisestä vallitsee ehdoton tietämättömyys,

rotusota, joka pakottaa koko Europan alistumaan viidentoista tai
parinkymmenen miljoonan aseistetun miehen hävitettäväksi ja joka
jo nyt ainoastaan siitä syystä ei riehu, että kaikista sotilasvaltioista
vahvintakin pelottaa lopputuloksen täydellinen ennakolta laskemisen
mahdottomuus.
Sitä suurempi on velvollisuus vetää 1870 vuoden kansainvälisen
työväenpolitiikan kaukonäköisyyden puoleksi unohduksiin jääneet
loistavat todistukset esiin saksalaisen työväestön nähtäviksi. Se mitä
on sanottu näistä kahdesta pääneuvoston selostuksesta koskee myös
"Kansalaissotaa Ranskassa". Toukokuun 28 p:nä sortuivat viimeiset
kommunitaistelijat Bellevillen rinteillä ylivoiman alle ja jo kaksi päivää
myöhemmin, 30:nä, luki Marx pääneuvostolle tämän teoksen, jossa
Pariisin kommunin historiallinen merkitys on esitetty lyhyin,
voimakkain, mutta niin terävin ja ennen kaikkea todellisin piirtein,
että vertaista ei ole löydettävissä koko asiata käsittelevästä
runsaasta kirjallisuudesta.
Ranskassa v:n 1789 jälkeen tapahtuneen taloudellisen ja
valtiollisen kehityksen vaikutuksesta oli Pariisi viitisenkymmentä
vuotta sitten tullut sellaiseen asemaan, että siellä ei saattanut
puhjeta minkäänlaista vallankumousta, jolla olisi ollut muu kuin
proletarinen luonne, niin että köyhälistö, ostettuaan voiton verellään,
esiintyi omin vaatimuksin. Nämä vaatimukset ovat olleet enemmän
tai vähemmän sekavia, vastaten Pariisin työläisten kehitysastetta
kulloinkin kysymyksessä olevana ajankohtana. Mutta lopuksikin
tarkoittivat ne aina luokkavastakohtien poistamista kapitalistien ja
työmiesten väliltä. Miten sen piti tapahtua, sitä ei tosiaankaan
tiedetty. Mutta vaatimus itsessään, niin epämääräinen kuin se olikin,
sisälsi vaaran vallitsevalle yhteiskuntajärjestykselle; työläiset, jotka
sen asettivat, olivat vielä aseissa; valtion ohjaksissa olevalle

porvaristolle oli siitä syystä työläisten riisuminen aseista ensimäinen
käsky. Sentähden jokaisen työläisten voittaman vallankumouksen
jälkeen syttyy uusi taistelu, joka päättyy työläisten häviöön.
Se tapahtui ensi kerran 1848. Parlamentarisen vastustuspuolueen
vapaamieliset porvarit toimeenpanivat juhlapäivällisiä ajaakseen läpi
vaaliuudistuksen, jonka piti taata niiden puolueelle valta. Ollen
taistelussa hallitusta vastaan pakotettuja yhä enemmän ja enemmän
vetoamaan kansaan, täytyi heidän antaa yläluokan ja
pikkuporvariston radikalisille ja tasavaltaisille kerroksille etusija.
Mutta näiden takana olivat vallankumoukselliset työläiset, ja nämä
olivat vuoden 1830 jälkeen saavuttaneet paljon suuremman
valtiollisen itsenäisyyden kuin porvarit ja tasavaltalaisetkaan
aavistivat. Hallituksen ja vastustuspuolueen välisen kiistan
kireimmällä hetkellä alkoivat työläiset katutaistelun. Louis Philipp
katosi, vaaliuudistus hänen mukanaan, niiden tilalle syntyi tasavalta,
vieläpä voitokkaiden työmiesten itsensä "yhteiskunnalliseksi"
kutsuma tasavalta. Mitä tällä yhteiskunnallisella tasavallalla oli
ymmärrettävä, siitä ei kukaan ollut selvillä, eivät työläiset itsekään.
Mutta heillä oli nyt aseita ja olivat yhtenä voimana valtiossa.
Sentähden, niin pian kuin peräsimessä istuvat porvaristasavaltalaiset
alkoivat tuntea saaneensa jotenkuten vankan pohjan jalkojensa alle,
oli heidän lähin päämääränsä riisua työläiset aseista. Tämä tapahtui
siten, että suoranaisten valeiden ja julkisen pilkan avulla sekä
koettamalla karkoittaa työttömät erääseen syrjäiseen maakuntaan,
pakotettiin työläiset nousemaan heinäkuun kapinaan 1848. Hallitus
oli pitänyt huolen musertavan ylivoiman saamisesta. Viisipäiväisen
sankarillisen taistelun jälkeen sortuivat työläiset. Ja nyt seurasi
turvattomien vankien joukossa verilöyly, jonka vertaista ei ole nähty
niiden kansalaissotien päivien jälkeen, jotka aloittivat Rooman
tasavallan häviön. Silloin porvaristo ensimäisen kerran näytti, mihin

mielettömiin julmuuksiin se voi kiihoittua kostossaan, niin pian kuin
köyhälistö itsenäisenä luokkana uskaltaa sitä vastaan esittää omia
vaatimuksiaan. Ja kuitenkin oli vuosi 1848 lastenleikkiä verrattuna
heidän raivoonsa vuonna 1871.
Rangaistus seurasi heti jälestä. Jollei köyhälistö kyennyt vielä
hallitsemaan Ranskaa, niin ei porvaristokaan pystynyt siihen enää. Ei
ainakaan silloin, kun sen enemmistö vielä oli yksinvaltaista
mieleltään ja jakaantuneena kolmeen dynastiseen (hallitsijasukua
kannattavaan) puolueeseen ja neljänteen tasavaltaiseen. Heidän
keskinäiset riitansa tekivät mahdolliseksi, että seikkailija Louis
Bonaparte saattoi ottaa haltuunsa kaikki valtapaikat — armeijan,
poliisin, hallintokoneiston — ja 2 p:nä joulukuuta hajoittaa
porvarillisten viimeisen lujan linnan, kansalliskokouksen. Toinen
keisarikunta alkoi, Ranska tuli valtiollisten seikkailijain ja
rahahuijarien joukkion nyljettäväksi, mutta samalla alkoi myös
teollinen kehitys, joka ei koskaan olisi ollut mahdollinen Louis
Philippin ahdasmielisen ja turhantarkan järjestelmän aikana, jolloin
kourallinen suurporvaristoa piti yksinään valtaa käsissään. Louis
Bonaparte otti kapitalisteilta valtiollisen vallan sillä tekosyyllä, että
suojelee porvaristoa työläisiä vastaan ja toiselta puolelta työläisiä
porvareita vastaan. Mutta siitä huolimatta suosi hänen valtansa
keinottelua ja teollista yritteliäisyyttä, lyhyesti sanoen, koko
porvariston nousua ja rikastumista ennen kuulumattomassa
määrässä. Tosin vielä paljon suuremmassa määrässä kehittyivät
lahjomiset ja joukkovarkaudet, jotka ryhmittyivät keisarillisen hovin
ympärille ja kiskoivat ansaituista rikkauksista vahvat prosenttinsa.
Mutta toinen keisarivalta oli ranskalaiselle kansallispöyhkeydelle
puhallettu yhteentoitotus, se oli 1814 menetettyjen ensimäisen
keisarikunnan, ainakin ensimäisen tasavallan rajojen takaisin

vaatimista. Ranskalainen keisarikunta oli vanhan yksinvallan rajojen
sisällä, vieläpä vuoden 1815 vielä enemmän leikeltyjen rajojen
puitteissa ajan pitkään mahdottomuus. Siitä johtui ajoittaisten sotien
ja rajanlaajennusten välttämättömyys. Mutta mikään rajanlaajennus
ei niin häikäissyt ranskalaisten kansallisylpeilijöiden mielikuvitusta
kuin saksalaisen vasemman Rheinin-rannan valloitus.
Neliöpeninkulma Rheinin rantaa merkitsi niille enemmän kuin
kymmenen Alpeilta tai jostakin muualta. Toiselle keisarikunnalle oli
vasemman Rheinin rannan takaisin vaatiminen, kokonaan kerrallaan
tai kappaleittain, ainoastaan hetkenkysymys. Tämä hetki tuli
preussiläis-itävaltalaisen sodan mukana 1866. Bismarckin ja oman
rikkiviisaan vitkastelupolitiikkansa pettämänä, mitä odoteltuun
"aluekorvaukseen" tuli, ei Bonapartella enää ollut muuta valittavana
kuin sota, joka puhkesi 1870 ja kiidätti hänet ensin Sedan'iin ja sieltä
Wilhelmshöheen.
Välttämätön seuraus oli 4 p:nä syyskuuta 1870 alkanut Pariisin
vallankumous. Keisarikunta lysähti kokoon kuin korttirakennus ja
uudelleen julistettiin tasavalta. Mutta vihollinen seisoi porttien
ulkopuolella. Keisarikunnan armeijat olivat joko toivottomasti
saarrettuina Metzissä tai vangittuina Saksassa. Tässä hädässä salli
kansa entisen lakiasäätävän laitoksen pariisilaisedustajien julistautua
"kansallisen puolustuksen hallitukseksi". Tämän sallittiin tapahtua
sitä kernaammin, kun nyt kaikki asekuntoiset pariisilaiset olivat
puolustusta varten astuneet kansalliskaartiin ja olivat aseissa, joten
työläiset nyt muodostivat siinä suuren enemmistön. Mutta jo pian
puhkesi ilmi vastakohta melkein yksinomaan porvaristosta
kokoonpannun hallituksen ja aseistetun köyhälistön välillä. 31 p:nä
lokakuuta valtasivat työväenpataljoonat kaupungintalon
väkirynnäköllä ja ottivat osan hallituksen jäsenistä vangiksi. Kavallus,
hallituksen suoranainen valapattoisuus ja muutamien

porvaripataljoonien väliintulo vapauttivat heidät jälleen, ja jottei
sisällinen kansalaissota olisi päässyt syttymään vieraan sotavoiman
piirittämässä kaupungissa, annettiin entisen hallituksen jäädä
toimeensa.
Vihdoinkin, 28 p:nä tammikuuta 1871 antautui nälänhädän
näännyttämä kaupunki. Mutta sotahistoriassa siihen saakka
ennenkuulumattomalla kunnialla. Linnoitukset luovutettiin,
ympärysmuuri riisuttiin aseista, linjaväen ja liikkuvan kaartin aseet
annettiin pois, ne itse otettiin sotavangeiksi. Mutta kansalliskaarti piti
aseensa ja kanuunansa ja taipui ainoastaan aselepoon voittajien
kanssa. Eivätkä edes nämäkään uskaltaneet voittokulussa kulkea
Pariisiin. Vain pienen, päälle päätteeksi osittain julkisen puiston
muodostaman kolkan Pariisia uskalsivat ne miehittää ja senkin
ainoastaan pariksi päiväksi! Ja tuonkin ajan piirittivät noita miehiä,
jotka itse olivat 131 pitkää päivää pitäneet Pariisia piirityksessä,
Pariisin aseistetut työmiehet, jotka tarkasti vartioivat, ettei yksikään
"preussiläinen" päässyt astumaan vieraalle valloittajalle määrätyn
nurkan ahtaiden rajojen ulkopuolelle. Sellaista kunnioitusta herättivät
Pariisin työmiehet sotajoukossa, jonka edessä keisarikunnan kaikki
armeijat olivat laskeneet aseensa, ja preussiläisten junkkarien, jotka
olivat tulleet paikalle kostaakseen ihan vallankumouksen liedellä,
täytyi jäädä kunnioittavasti seisomaan ja tervehtiä juuri tuota
aseistettua vallankumousta!
Sodan aikana olivat Pariisin työläiset rajoittuneet vaatimaan
taistelun tarmokasta jatkamista. Mutta nyt, kun Pariisin
antautumisen jälkeen tehtiin rauha, täytyi Thiers'in, hallituksen
uuden päämiehen, havaita, että omistavien luokkien — suurten
maanomistajien ja kapitalistien — valtaa uhkasi alituinen vaara niin
kauan kuin Pariisin työläisillä oli aseet käsissään. Hänen ensimäinen

tehtävänsä oli yrittää riisua ne aseista. 18 p:nä maaliskuuta lähetti
hän linja-joukot ryöstämään kansalliskaartille kuuluvat, Pariisin
piirityksen aikana valmistettua ja julkisella listakeräyksellä maksettua
tykistöä. Yritys epäonnistui, Pariisi nousi yhtenä miehenä
vastarintaan ja sota Pariisin ja Versailles'issa majailevan Ranskan
hallituksen välillä oli julistettu. Maaliskuun 26 p:nä valittiin Pariisiin
kommuuni ja julistettiin 28 p:nä. Kansalliskaartin keskuskomitea,
joka siihen asti oli hoitanut hallitusta, luovutti valtansa sille,
sittenkun se sitä ennen vielä oli antanut määräyksen Pariisin
häpeällisen "siveyspoliisin" lakkauttamisesta. 30 p:nä lakkautti
kommuuni sotaväenoton ja seisovan armeijan ja julisti
kansalliskaartin, johon kaikkien asekuntoisten kansalaisten tuli
kuulua, ainoaksi aseelliseksi voimaksi. Se antoi anteeksi kaikki
asuntovuokrat vuoden 1870 lokakuusta huhtikuuhun saakka,
laskemalla jo maksetut vuokrasummat siitä alkavan vuokra-ajan
maksuiksi, ja lakkautti kaiken panttienmyynnin kaupungin
panttilainastoissa. Samana päivänä vahvistettiin kommuuniin valitut
ulkomaalaiset virkoihinsa, sillä "kommuunin lippu on
maailmantasavallan lippu". — 1 p:nä huhtikuuta päätettiin, että
kommuunin palveluksessa olevan henkilön, siis myös kommuunin
jäsenten itsensä korkein palkka ei saanut nousta yli 6,000 frangin
(6,000 mk). Seuraavana päivänä määrättiin kirkko erotettavaksi
valtiosta ja kaikki kirkollisiin tarkoituksiin menevät valtionmaksut
lakkautettaviksi samoin kuin kaikki kirkolliset tilat muutettaviksi
kansallisomaisuudeksi. Sen johdosta käskettiin 8 p. huhtik. kouluista
julistettavaksi pannaan kaikki uskonnolliset vertauskuvat, kuvat,
uskonkappaleet, rukoukset, lyhyesti sanoen "kaikki mikä kuuluu
kunkin yksityisen omantunnon piiriin" ja toteutettiin vähitellen. — 5
p:nä annettiin sen vastapainoksi, että vanhan hallituksen joukot
päivittäin ampuivat vangittuja kommuunitaistelijoita, käsky pantiksi

otettujen henkilöiden vangitsemisesta, jota ei kuitenkaan pantu
täytäntöön. — 6 p:nä nouti kansalliskaartin 137:s pataljoona
giljotiinin, joka kansan äänekkäästi riemuitessa poltettiin julkisesti. —
12 p:nä päätti kommuuni syöstä alas 1809 v:n Napoleonin sodan
jälkeen Vendôme-torille pystytetyn, valloitetuista kanuunista valetun
voitonpatsaan kansallisylpeyttä ja kansojen toisiaan vastaan
kiihottamista muistuttavana. Se pantiin täytäntöön 16 p:nä
toukokuuta. — Samana päivänä päätettiin antaa laatia tilastollinen
selonteko tehtailijoiden seisauttamista tehtaista ja valmistaa
suunnitelmia näiden tehtaiden käyttämiseksi niissä työskennelleiden
työmiesten avulla; työläisten piti muodostaa osuustoiminnallisia
yhdistyksiä, jotka taas olivat liitettävät suuremmaksi liitoksi. — 20
p:nä lakkautettiin leipurien yötyö samaten kuin aina toisen
keisarikunnan päiviltä asti voimassa ollut, poliisin nimittämien
henkilöiden — ensiluokkaisten työriistäjien — monopoolina
(yksinoikeutena) harjoittama työnvälitys, joka annettiin Pariisin
kahdenkymmenen eri piirin (arrondissment) määrien (maire)
huoleksi. — 30 p:nä huhtikuuta määräsi kommuuni lakkautettavaksi
panttilainakonttorit, jotka olivat, kuten sanottiin, työläisten yksityistä
nylkemistä varten ja olivat ristiriidassa sen oikeuden kanssa, mikä
työläisillä oli työkaluihinsa ja luottoon. — 5 p:nä toukok. päätti se
hajoittaa Ludvig XVI:nnen mestauksen sovitukseksi rakennetun
rukous-kappelin.
Niin jyrkkänä ja puhtaana ilmeni maalisk. 18 p:n jälkeen
pariisilaisen liikkeen luokkaluonne, jonka taistelu maahan karannutta
vihollista vastaan oli tunkenut siihen asti syrjään. Kun kommuunissa
istui melkein yksinomaan työläisiä tai tunnettuja työväenedustajia,
niin oli myös sen päätöksillä ilmeisesti proletarinen luonne. Joko
määräsi se toimeenpantavaksi uudistuksia, jotka tasavaltalainen
porvaristo yksistään pelkuruudesta oli laiminlyönyt, mutta jotka

muodostivat välttämättömän perustuksen työväenluokan vapaalle
toiminnalle, kuten esim. sen lauseen toteuttaminen, että uskonto
suhteessaan valtioon on yksityinen asia; tai antoi se päätöksiä, jotka
olivat suorastaan työväenluokan eduksi tehtyjä ja jotka koskivat
osittain syvästi vanhaan yhteiskuntajärjestykseen. Mutta kaikkien
näiden toteuttaminen voitiin piiritetyssä kaupungissa korkeintaan
vasta panna alulle. Ja toukokuun alusta alkaen vaati taistelu
Versailles'n hallituksen yhä lukuisammiksi karttuvia joukkoja vastaan
kaikki voimat.
7 p:nä huhtikuuta olivat versaillesilaiset vallanneet itselleen
ylipääsyn Seinen yli Neuillyn luona Pariisin länsirintamalla; sitävastoin
löi kenraali Eudes 11 p. takaisin niiden verisen hyökkäyksen eteläistä
rintamaa vastaan. Pariisia pommitettiin yhteen menoon, ja sitä
tekivät samat ihmiset, jotka olivat leimanneet preussilaisten
pommituksen samaa kaupunkia vastaan pyhyyden loukkaukseksi.
Nämä samat ihmiset kerjäsivät nyt Preussin hallitukselta, että se
lähettäisi pikaisesti Sedan'ista ja Metzista vangitut ranskalaiset
sotamiehet valloittamaan Pariisin heille takaisin. Näiden joukkojen
vähittäinen saapuminen antoi toukokuun alusta versaillesilaisille
ratkaisevan ylivoiman. Tämä osoittautui jo siinä, että Thiers 23 p:nä
katkaisi keskustelut, jotka koskivat sellaista kommuunin tarjoamaa
vaihtoa, että Pariisin arkkipiispa ja koko joukko muita Pariisissa
panttivankeina pidettyjä pappeja olisi vaihdettu yksinään Blanqui'ta
vastaan, joka kahdesti oli valittu kommuuniin, mutta oli Clairvaux'ssa
vankina. Mutta vielä enemmän ilmeni se Thiers'in muuttuneessa
kielenkäytössä; oltuaan tähän asti pidättyväinen ja kaksikielinen,
muuttui hän äkkiä hävyttömäksi, röyhkeäksi, raa'aksi. Etelärintamalla
ottivat versaillesilaiset 3 p:nä toukokuuta Moulin Saquet'n
kenttävarustukset, 9 p:nä täydellisesti mäsäksi ammutun Issy'n
linnoituksen ja 14 p:nä Vanves'in linnan. Länsirintamalla hyökkäsivät

he vähitellen, lukuisia ympärysmuuriin saakka ulottuvia kyliä ja
rakennuksia valloitellen, ihan päävallin luokse asti. 11 p:nä onnistui
heidän kavalluksen ja sinne asetetun kansalliskaartin
huolimattomuuden tähden tunkeutua kaupunkiin. Preussiläiset, jotka
pitivät miehitettyinä pohjoisia ja itäisiä linnoituksia, sallivat
versaillesilaisten tunkeutua eteenpäin yli heiltä aselevossa kielletyn
alueen kaupungin pohjoisosassa ja senkautta ryhtyä hyökkäämään
sillä pitkällä rintamalla, jonka pariisilaiset luulivat olevan aselevolla
suojatun ja jota siitä syystä pitivät heikosti miehitettynä. Tämän
johdosta oli vastarinta Pariisin länsiosassa, varsinaisessa
loistokaupunginosassa, ainoastaan heikkoa; se muuttui sitä
ankarammaksi ja sitkeämmäksi, mitä lähemmäksi itäistä puolta,
varsinaista työväenkaupunginosaa, eteenpäin tunkeutuvat joukot
tulivat. Vasta kahdeksanpäiväisen taistelun jälkeen kukistuivat
viimeiset kommuunin puolustajat Bellevillen ja Menilmontant'in
kukkuloilla, ja nyt saavutti turvattomien miesten, naisten ja lasten
murhaaminen, joka yltyen oli raivonnut läpi koko viikon, huippunsa.
Takaaladattava ei enää tappanut kyllin nopeasti, sadottain voitettuja
ammuttiin kuularuiskuilla mäsäksi. "Liittoutuneiden muuri" Père
Lachaisen kirkkopihalla, jossa viimeinen joukkomurha
toimeenpantiin, seisoo vielä tänäpäivänä kaikessa mykkyydessään
paljon puhuvana todistuksena siitä raivosta, mihin hallitseva luokka
on valmis niin pian kuin köyhälistö uskaltaa nousta esiintymään
oikeuksiensa puolesta. Sitten seurasivat joukkovangitsemiset, kun
kaikkien teurastaminen osoittautui mahdottomaksi, vangittujen
riveistä mielivaltaisesti poimittujen teurasuhrien ampumiset, loppujen
kulettaminen suuriin leireihin, joissa he odottivat raahaamistaan
sotaoikeuksien tuomittaviksi. Preussiläisiä joukkoja, jotka piirittivät
Pariisin koillisosaa, oli kielletty laskemasta lävitsensä yhtään
pakolaista, mutta kuitenkin sulkivat upseerit usein silmänsä, kun

havaitsivat sotamiehen enemmän noudattavan ihmisyyden kuin
ylipäällikön käskyä; erityisesti ansaitsee saksilainen armeijakunta
tulla mainituksi siitä ihmisystävällisestä menettelystään, että se laski
läpi useita, joiden osallisuus kommuunitaisteluihin oli ilmeinen.
* * * * *
Jos tänään, kahdenkymmenen vuoden kuluttua, tarkastelemme
v:n 1871 Pariisin kommuunin toimintaa ja historiallista merkitystä,
niin tulemme havaitsemaan, että "Kansalaissodassa" annettuun
esitykseen on vielä tehtävä muutamia lisäyksiä.
Kommuunin jäsenet olivat jakaantuneina enemmistöön,
blanquisteihin, jotka myöskin olivat olleet vallalla kansalliskaartin
keskuskomiteassa, ja vähemmistöön, jonka etupäässä muodostivat
Proudhon'in sosialistista suuntaa kannattavat kansainvälisen
työväenpuolueen jäsenet. Blanquistit olivat silloin suurelta osaltaan
sosialisteja ainoastaan vallankumouksellisesta, proletarisesta
vaistosta; ainoastaan muutamat harvat olivat Vaillant'in kautta, joka
tunsi saksalaista tieteellistä sosialismia, päässeet suurempaan
periaatteelliseen selvyyteen. Niin on käsitettävissä, että
taloudellisessa suhteessa lyötiin laimin paljon sellaista, mitä
kommuunin meidän nykyisen katsantokantamme mukaan olisi
pitänyt tehdä. Tosin kaikkein vaikeimmin ymmärrettävissä on se
pyhä kunnioitus, jolla nöyrinä jäätiin seisomaan Ranskan pankin
porttien ulkopuolelle. Se oli myöskin raskas poliittinen virhe. Pankki
kommuunin käsissä — olisi ollut suuremman arvoinen kuin
kymmenentuhatta panttivankia. Se olisi vaikuttanut, että koko
Ranskan porvaristo olisi painostanut Versaillesin hallitusta tekemään
rauhan kommuunin kanssa. Mutta vielä ihmeteltävämpää on se suuri
määrä oikeata, mitä blanquisteista ja proudhonilaisista kokoonpantu

kommuuni siitä huolimatta teki. Luonnollisesti ovat ensi kädessä
proudhonilaiset vastuunalaisia kommuunin taloudellisista päätöksistä,
niiden sekä kiitettävistä että moitittavista puolista, ja blanquistit
vastuunalaisia sen poliittisista teoista ja tekemättä jättämisistä. Ja
kummassakin tapauksessa tahtoi historian iva — kuten tavallisesti,
kun tieteilijät pääsevät peräsimeen käsiksi —, että niin toiset kuin
toisetkin tekivät ihan päinvastoin kuin heidän koulukuntansa oppi
heille määräsi.
Proudhon, pikkutilallisten ja käsityöläismestarien sosialisti, vihasi
yhdyskunnaksi liittymistä positivisella vihalla. Hän sanoi sen tuovan
enemmän pahaa kuin hyvää, olevan luonnostaan hedelmätöntä, jopa
vahingollista, koska se kahlehtii työläisten vapautta; se oli hänen
mielestään pelkkä uskonkappale, hyödytön ja ehkäisevä, ristiriidassa
niin hyvin työläisten vapauden kuin työn säästämisen kanssa, ja sen
varjopuolet kasvavat nopeammin kuin sen edut; sitä vastoin olivat
kilpailu, työnjako, yksityisomaisuus taloudellisia voimia. Ainoastaan
poikkeustapauksissa, kuten Proudhon niitä nimittää, —
suurteollisuudessa ja suurissa liikeyrityksissä, esim. rautateillä — oli
työläisten yhteenliittyminen paikallaan. (Ktso Idée général de la
révolution, 3. étude.)
Jo 1871 oli suurteollisuus itse Pariisissa, taiteellisen käsityön
pääpaikassa, siinä määrin lakannut olemasta poikkeustapaus, että
kommuunin verrattomasti tärkein päätös määräsi suur- ja vieläpä
käsityöteollisuudelle järjestelyn, joka ei ainoastaan perustunut
työläisten yhteenliittymiseen, vaan jonka myös piti yhdistää nämä
liittymät yhdeksi suureksi liitoksi, lyhyesti sanottuna järjestelmän,
jonka, kuten Marx "Kansalaissodassaan" aivan oikein huomauttaa,
loppujen lopuksi täytyisi johtaa kommunismiin, siis aivan
vastakkaiseen suuntaan kuin Proudhon'in oppi. Ja siitä syystä olikin

kommuuni proudhonilaisen sosialistikoulun hauta. Tämä koulu onkin
nykyään hävinnyt ranskalaisista työväenpiireistä. Siellä on nyt niin
possibilistien kuin marxilaistenkin keskuudessa kieltämättömästi
vallalla marxilainen teoria. Ainoastaan "radikalisten" porvarien
joukossa on vielä proudhonilaisia.
Blanquisteille ei käynyt paremmin. Salaliittojen kouluissa
kasvaneina ja niissä käytettävän ankaran kurin koossapitäminä oli
niillä lähtökohtana mielipide, että suhteellisesti pieni määrä
päättäväisiä, hyvin järjestyneitä miehiä pystyy suotuisan hetken
tultua ei ainoastaan tarttumaan valtion ohjaksiin vaan myöskin
suurta ja häikäilemätöntä tarmoa käyttämällä pitämään ne
käsissään, siksi kunnes ovat onnistuneet tempaamaan kansanjoukot
vallankumouksen pyörteeseen ja keräämään ne pienen
johtajajoukon ympärille. Siihen kuului ennen kaikkea kaiken vallan
mitä ankarin, diktaattorimainen keskittäminen uuden
vallankumouksellisen hallituksen käsiin. Ja mitä teki kommuuni,
jonka enemmistönä olivat juuri nämä blanquistit? Kaikissa
julistuksissaan maaseudun ranskalaisille kehoitti se näitä kaikkien
kuntien vapaaseen liittoon Pariisin kanssa, kansalliseksi järjestöksi,
jonka kansakunta nyt ensi kerran itse loisi. Juuri tähänastisen
hallituksen keskitetyn sortovallan armeijoineen, valtiollisine
poliiseineen ja virkavaltoineen, jonka Napoleon oli 1798 luonut ja
jonka jokainen uusi hallitus oli siitä lähtien ottanut tervetulleena
aseena vastaan ja käyttänyt vastustajiaan vastaan, juuri tämän
vallan piti kaikkialla kaatua, niinkuin se jo oli Pariisissa kukistunut.
Kommuunin täytyi heti alussa tunnustaa, että työväenluokka,
kerran valtaan päässeenä, ei enää kauemmin voinut hoitaa taloutta
vanhan valtiokoneiston avulla; että tämän saman luokan, jottei se
menettäisi takaisin omaa, äsken valloittamaansa valtaa, täytyi

toiselta puolen hävittää koko vanha, siihen asti sitä itseään vastaan
käytetty sortokoneisto, toiselta puolelta turvata itsensä omia virka- ja
valtiopäivämiehiään vastaan julistamalla, että ne voitiin
poikkeuksettomasti ja milloin tahansa erottaa. Missä ilmeni
tähänastisen valtion luonteenomainen omaisuus? Yhteiskunta oli
alkuaan yhteisten etujensa huoltamista varten luonut yksinkertaisen
työnjaon kautta itselleen omia orgaaneja, elimiä. Mutta nämä
orgaanit, joiden huippuna on valtiovalta, olivat aikaa yhteisen, omien
erikoisetujensa palveluksessa, muuttuneet yhdyskunnan palvelijoista
sen herroiksi. Tämä on havaittavissa niin hyvin demokraattisissa
tasavalloissa kuin perinnöllisissä yksinvalloissakin. Missään eivät
"valtiomiehet" muodosta eristetympää ja mahtavampaa osaa
kansasta kuin juuri Pohjois-Amerikassa. Siellä hallitsevat kumpaakin
niistä kahdesta suuresta puolueesta, jotka vaihdellen ovat vallassa,
vuorostaan ihmiset, jotka harjoittavat politiikkaa hyödyksensä, jotka
keinottelevat itselleen paikkoja liittovaltion ja yksityisten valtioitten
lakiasäätävissä laitoksissa tai jotka elävät puolueensa
vaaliyllytyksestä ja puolueen voitettua vaaleissa saavat hyviä virkoja
palkaksensa. Tiedetään, miten amerikalaiset ovat 30 vuotta
koittaneet ravistaa tätä sietämättömäksi käynyttä iestä niskastaan,
mutta kaikesta huolimatta vajoavat yhä syvemmälle tähän
turmeluksen ja lahjomisien suohon. Juuri Amerikassa voimme
paraiten nähdä, miten tämä valtiovallan vieraantuminen
yhteiskunnasta, jonka yksinomaiseksi välikappaleeksi se alkuaan oli
tarkoitettu, tapahtuu. Siellä ei ole olemassa hallitsijasukua, ei
aatelistoa, ja seisovata sotaväkeä, lukuunottamatta sitä pientä
joukkoa, joka on intiaanien vartioimista varten, ei ole virkavaltaa
vakinaisine virkoineen ja eläkkeensaamisoikeuksineen. Ja kuitenkin
on siellä kaksi joukkuetta valtiollisia keinottelijoita, jotka vaihdellen
pitävät valtiovaltaa ja käyttävät mitä turmeltuneimpia keinoja

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Rational Function Systems And Electrical Networks With Multiparameters Kaisheng Lu

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