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1. ME 242
Applied Mathematics for
Engineers
Linear Algebra –V
(Orthogonal and Similar matrices,
Similarity transformation, Diagonalization)
Prof.Dr. H. SeçilARTEM

2. ME242 Applied Mathematics for Engineers
A real square matrix [A] is called
Symmetricif transposition leaves it unchanged
Skew-symmetriciftranspositiongivesthenegativeof[A]
[A]
T =[A]
[A]
T = -[A]
Symmetric, Skew-symmetric and Orthogonal Matrices
Orthogonaliftranspositiongivestheinverseof[A] [A]
T =[A]
-1
•The eigenvaluesofarealsymmetricmatrixarereal
•The eigenvaluesofarealskew-symmetricmatrixarepureimaginaryorzero
•The eigenvalues of an orthogonal matrix are real or complex conjugates in pairs and have
absolute value 1.

3. ME242 Applied Mathematics for Engineers
Orthogonal Transformations are transformations
[y]=[A][x] where [A]is an orthogonal matrix
Ex: Plane rotation through an angle is an orthogonal transformation
y=
cos−sin
sincos
x
1
x
2
Invariance of Inner Product
An orthogonal transformation preserves the value of the inner product of vectors [a] and [b] in ??????
??????
,
defined by
[a] . [b] =[�]
??????
[�]= [�
1….�
??????]
�
1
⋮
�
??????
An orthogonal transformation also preserves the length or norm of any vector [a] in ??????
??????
[�]= �.[�]=[�]
??????
[�]

4. ME242 Applied Mathematics for Engineers
Orthonormality of column and row vectors
A real square matrix is orthogonal if and only if its column vectors
[�]
1,…..,[�]
??????(and also its row vectors) form an orthonormal system, that is,
[�]
�.[�]
�= [�]
�
??????
[�]
�=
0�??????�≠�
1�??????�=�
•The determinant of an orthogonal matrix has the value +1 or -1
•The eigenvalues of an orthogonal matrix are real or complex conjugates in pairs
and have absolute value 1.
??????=
2/31/32/3
−2/32/31/3
1/32/3−2/3
=[�]
1[�]
2[�]
3
Ex:Is the following matrix orthogonal?
[�]
1.[�]
2=2/3−2/31/3
1/3
2/3
2/3
= 0
All possible product of vectors are satisfied;
hence, [A] matrix is an orthogonal matrix

5. SimilarMatrices,SimilarityTransformationandDiagonalization
Twosquarematrices[A]and[B]ofthesamesizenarecalledsimilar
ifthereexitsanon-singularmatrix[P]ofsizensuchthat
thematrix[B]isobtainedbymeansofaso-calledsimilaritytransformationdefinedas
[B]=[P]
-1 [A][P]or[P][B]=[A][P]or[A]=[P][B][P]
-1
oralternatively
[B]=[Q][A][Q]
-1
or[B][Q]=[Q][A]or[A]=[Q]
-1 [B][Q]
where
[Q]=[P]
-1
ME242 Applied Mathematics for Engineers

6. Ex:Determinewhetherthefollowing[A]and [B]matricesare similarornot.
The similaritytransformation requiresthat[P][B]=[A][P]
Define[P]as
p
112p
122p
21
p
112p
212p
22
2p
122p
22
p
12p
22
00

=
00

12 20
 
14 23
[A] [B]

p
12
p
22
[P]
p
21
p
11
[A][P]
p
22
2p
11
4p
21
p
12
31
01p
122
p
222p
21
p
11
[P][B] 


ME242 Applied Mathematics for Engineers

7. Thisgivesthese fourequationsinfourunknowns
p
11
p
11
-2p
12-2p
21=0
-2p
21-2p
22= 0
2p
12-2p
22= 0
p
12-p
22=0
Define,arbitrarily,p
22=c
1andp
21=c
2,then
[P]becomesnon-singularwhen c
10 and 2c
1–c
2.
Hence,[A]and [B]matricesaresimilar.

2c
12c
2c
1
c
1
[P]
c
2
whose determinantisfound asdet[P]=c
1(2c
1+c
2)
ME242 Applied Mathematics for Engineers

8. Ex: Determinewhetherthefollowing[A]and [B]matricesare similarornot.
ME242 Applied Mathematics for Engineers
The similaritytransformation requiresthat [P][B]=[A][P]
whose solution isfound asp
22 =0,p
12 =0,p
11 =p
21 =c≠0to give
[P]issingular (since det[P]=0)
whatevercis.
Hence,[A]and [B]matricesarenot
similar.

9. Property 2:
Similar matrices
have thesame characteristicpolynomial
•thesame characteristicequation
•thesame eigenvalues
•thesame trace
•thesame determinant
•thesame rank
*Notethattheconverse maynotalwaysbe true
ME242 Applied Mathematics for Engineers
Some propertiesofmatrix similarity
Property 1:Matrixsimilarityisan equivalencerelation
•Anysquare matrix[A]issimilartoitself Reflexive
•Ifthematrix[B]issimilarto a matrix[A],then[A]issimilarto[B]Symmetric
•If[C]issimilarto[B]and [B]to[A],then[C]issimilarto [A] Transitive
Thispropertymayalso bestatedas:The characteristic
polynomial,characteristic equation,eigenvalues,trace,
determinant,and rankofamatrixisinvariantunder
similaritytransformation.

10. Theorem:Considertwomatrices[A]and[B]ofthesamesizenwithn-distinct
eigenvalues;iftheireigenvaluesarethesameset,thenthesetwomatricesare
similartoeachother,otherwisetheyarenotsimilar.
Ex:Determinewhetherthefollowing[A]and[B]matricesaresimilarornot.
Since [A]and[B]have thesame eigenvalues,λ
1=2andλ
2=3,theyaresimilar.
det{[A]-λ[I]}=
1λ2
14λ
2λ0
23λ
=(1-λ)(4-λ)-(2)(-1)=λ
2-5λ+6=(λ-2)(λ-3)
det{[B]-λ[I]}= =(2-λ)(3-λ)-(-2)(0)=(λ-2)(λ-3)
12 20
 
14
[A]
23
[B]
ME242 Applied Mathematics for Engineers

11. Ex: Determinetheeigenvectorsofthefollowing matrices:
Notethat[A]and [B]aresimilarwithcommon eigenvaluesλ
1=2andλ
2=3.
Theeigenvectorsof[A]canbe foundas
Since the
transformation
matrixbetween
[A]and [B]is
known as
12 20
 
14 23
[A] [B]
[Aλ
1[I[x
1 [x
1
1
2


2x
210
x1 2x10

142x211
122 1

[Aλ
2[I[x
2 [x
2
1
1

1x
220
2x10

143x221
The eigenvectors
of[B]arefound as
2

x1132
1

11
[P]
0

11
10

1
-1
11
[P]
-1

0

012
121

1

1
[y][P][x]
1 1
-1

0 11
0111
[y]
2[P]
-1
[x]
2
1
Property 3:
If[x]isaneigenvectorofamatrix[A]and,furthermore,[B]=[P]
-1 [A][P],then[P]
-1 [x]isan
eigenvectorofmatrix[B].

12. Diagonalization
Ofspecialinterestamongvariousformsofsimilarmatricesisthediagonalform
whichhasitseigenvalueslocatedinthemaindiagonal whilealloffdiagonal
elementsarezero.
Inthiscase,the specialtransformationmatrix[T]toyield[D]=[T]
-1[A][T]isgivenby
[T]={[x]
1¦[x]
2¦…¦[x]
n}
where[x]
i isthe i
theigenvectorof[A]associated withitsi
theigenvalue 
i.
[D
0


λ
100

λ
20



00λ
n


ME242 Applied Mathematics for Engineers
[D

13. Whenthematrix[A]possessesdistincteigenvalues,sincealleigenvectorswillbe
linearlyindependent,theresultingtransformationmatrix[T]hastobenon-singular,
whichwillensureaproperdiagonalizationprocess.
Ex:Diagonalizethefollowingmatrix:
Apropertransformationmatrixis
-1/37/6

0-1/

2


-1/31/31/3



1/2
-1/6

1

3

131

02

11
[T]
1



1

3

131
2
1
1
[x][x]}

0

[T]{[x]
1
2 31
Diagonalization gives

1

1

112
[A]

12
1


0
Recallthat
λ
1=-1,λ
2=1,λ
3=2

3


1



1


1



2

3

0


1

[x]

1
,[x],[x]
321
32100
2 1

12002
0


6



0

1/3-1
-1/2

0
-1/37/6-1
0
1/3
1-1/3

12131-1/6
2 23



1/2
1 111

0
1
2

1/30
-1/

1
-1/37/61

0
-1/31/3
[T]
1
[A][T]

1/2
-1/6
ME242 Applied Mathematics for Engineers
( [ D]=[T]
-1[A][T])

14. Ex: Determine[A]
5and [A]
-1forthe followingmatrix:
[A=
12

-1
4

 
Theorem:Forsquarematrix[A],ifa transformationmatrix[T]existssuchthat
If[A]isnon-singular,the resultisalso true foranynegativeinteger;inparticular
Recallthatλ
1=2,λ
2=3and 
1
1
,[x]

1
2
[x]
21
[A
5
[T[D
5
[T
1

2132011179422
   
45422112431110

2120
5
21
1
1

11

03
1
[A
1
[T[D
1
[T
1
 
1/3
1/6

21/6
0112/3

1/31
211/2

110

0
1
21
1
1

31

212
110
ME242 Applied Mathematics for Engineers
[D]=[T]
-1[A][T] then
[A
1
[T[D
1
[T
1
[A
k
[T[D
k
[T
1
for any positive k