Lecture-4 Reduction of Quadratic Form.pdf

Ouadratic form

1. Reduction Of
Quadratic Form
To
Canonical Form
1

2. QuadraticForm
2
Definition: The quadratic form in n variables �
1,�
2.,….�
??????is the general homogenous
function of second degree in the variables
i.e. Y=�(�
1,�
2.,….�
??????)=σ
�,�=1
??????
�
���
��
�
In terms of matrix notation , the quadratic form is given by
Y=�
??????
��=�
1�
2…�
??????
�
11�
12…�
1??????
�
21�
22…�
2??????
…………
�
??????1�
??????2…�
????????????
�
1
�
2
…
�
??????

3. InMatrixNotation
3
In two variables (�,�): ��
2
+2ℎ��+��
2
��,�,�=�
??????
��=��
�ℎ
ℎ�
�
�
Here A is known as matrix of quadratic form
.

4. Inthreevariables
4
�(�,�,�)=��
2
+��
2
+��
2
+2ℎ��+2���+2���
In matrix notation
��,�,�=�
??????
��=���
�ℎ�
ℎ��
���
�
�
�

5. Towritematrixofquadraticforminthreevariables
5
��,�,�=��
2
+��
2
+��
2
+2ℎ��+2���+2���
A=
�����??????�??????���(�
2
)
1
2
�����??????�??????���(��)
1
2
�����??????�??????���(��)
1
2
�����??????�??????���(��)�����??????�??????���(�
2
)
1
2
�����??????�??????���(��)
1
2
�����??????�??????���(��)
1
2
�����??????�??????���(��)�����??????�??????���(�
2
)

6. 6
Ex. Obtain the matrix of the Quadratic form
Q=�
2
+2�
2
−7�
2
−4��+8��+5��
Sol. Compare the given quadratic form by standard form
��,�,�=��
2
+��
2
+��
2
+2ℎ��+2���+2���
Here �=1,�=2,�=−7;ℎ=−2;�=4;�=
5
2
Thereforecoefficientmatrixis
�=
�ℎ�
ℎ��
���
=
1−24
−22
5
2
4
5
2
−7
Whichisasymmetricmatrix.

7. 7
Ex. Obtain the matrix of the Quadratic form
Q=2�
2
+3�
2
+�
2
−3��+2��+4��
�=
�ℎ�
ℎ��
���
=
2−
3
2
1
−
3
2
32
121
Ans.

8. �=
EX.Writedownthequadraticformcorrespondingtothe
matrix
�=
125
203
534
���.Quadraticform??????��
??????
���ℎ����=
�
�
�
�
??????
��=���
125
203
534
�
�
�
or ��,�,�=�
2
+0�
2
+4�
2
+4��+6��+10��
is the required Quadratic form.
8

9. Sol.Quadraticform=�
??????
���ℎ���X=
�
�
�
�
??????
��=���
324
204
443
�
�
�
or ��,�,�=3�
2
+0�
2
+3�
2
+4��+8��+8��
is the required Quadratic form.
EX.Writedownthequadraticformcorrespondingtothematrix
�=
324
204
443
9

10. CANONICALFORMOR
SUMOFSQUAREFORM
10
The sum of square form of a real quadratic form �
??????
��is
??????
??????
????????????=??????
�??????
�
�
+ ??????
�??????
�
�
+…… +??????
????????????
??????
�
is known as canonical form which is formed with the help of the
orthogonal transformation �=��where B is the modal matrixof
normalizedeigenvectors,Y=
�
1
�
2
…
�
??????
and D is a diagonal matrix whose
diagonal elements are the eigen values of the matrix A.

11. 11
1.Firstwritethecoefficientmatrixofthegivenquadratic
form.
2.Findtheeigenvaluesandeigenvectorofthecoefficient
matrixA.
3.Nowcheckwhethereigenvectorsarepairwise
orthogonalornot.
OrthogonalVector;Twovectors�
1,�
2aresaidtobe
orthogonaliftheirinnerproductiszeroi.e.�
1
??????
.�
2=0
Working Method

12. 12
4.FormamodalmatrixBofnormalizedeigen
vector
i.e.B=�
1 �
2 �
3
IfX
1=
�
1
�
2
�
3
then�
1=
�
1
�
1
2
+�
2
2
+�
3
2
�
2
�
1
2
+�
2
2
+�
3
2
�
3
�
1
2
+�
2
2
+�
3
2
Similarlyfor�
2and�
3
HereBismatrixoftransformation

13. 13
�
5. Here B matrix is an orthogonal matrix
∴��
??????
=??????
∵�
??????
=�
−1
Find �
−1
��
�
−1
��=??????(diagonal matrix whose diagonal elements are
eigen values of the matrix A)
The required canonical form is
�
??????
(�
−1
��)�= �
??????
??????�=�
1�
2�
3
??????
100
0??????
20
00??????
3
�
1
�
2
�
3
i.e. ??????
1�
1
2
+ ??????
2�
2
2
+ ??????
3�
3
2
is the required canonical form and
�=��isrequiredorthogonaltransformation.

14. Ex. Reduce the quadratic form
Q(�,�,�)=�
1
2
+3�
2
2
+3�
3
2
−2�
2�
3into canonical
form using orthogonal transformation.
Sol.Writethecoefficientmatrixofgivenquadratic
form
�=
100
03−1
0−13
TheeigenvaluesofAare1,2,4
14

15. EigenVectorsfor??????=1
�−??????
1??????�
1=����−??????�
1=�
or
000
02−1
0−12
�
1
�
2
�
3
=
0
0
0
or
000
02−1
00
3
2
�
1
�
2
�
3
=
0
0
0
Applying�
3→�
3+
1
2
�
2
2�
2−�
3=0.......(1)
�
3=0.............(2)
Hereno.ofequations(m)are2andno.of
variables(n)are3
n−m=3−2=1variableisindependent.
Let�
1=k
�
3=0;�
2=0
�
1=�
15

16. �
1=
�
1
�
2
�
3
=
�
0
0
If�=1�ℎ���
1
=
1
0
0
istheeigenvector
correspondingto??????=1
16

17. Eigenvectorfor??????=2
�−??????
2??????�
2=�or�−2??????�
2=�
−100
01−1
0−11
�
1
�
2
�
3
=
0
0
0
Applying�
3→�
3+�
2
−100
01−1
000
�
1
�
2
�
3
=
0
0
0
�
1=0.......(1)
�
2−�
3=0.............(2)
Hereno.ofequations(m)are2andno.of
variables(n)are3
n−m=3−2=1variableisindependent.
Let�
3=k
�
2=�
3,
�
3=�=�
2
17

18. �
2=
�
1
�
2
�
3
=
0
�
�
If�=1�ℎ���
2=
0
1
1
istheeigenvector
correspondingto??????=2
18

19. EigenVectorfor??????=4
�−??????
3??????�
3=�or�−4??????�
3=�
or
−300
0−1−1
0−1−1
�
1
�
2
�
3
=
0
0
0
�����??????��R
3→R
3−R
2
−300
0−1−1
000
�
1
�
2
�
3
=
0
0
0
�
1=0;�
2+�
3=0
Hereno.ofequationsare(m)2and
no.ofvariables(n)are3
n−m=3−2=1variableisindependent.
∴Let�
3=k
�
2=−�
3
�
2=−�;�
1=0
19

20. �
3=
�
1
�
2
�
3
=
0
−�
�
??????��=1�ℎ���
3=−
0
1
1
istheeigenvector
correspondingto??????=4
20

21. Continued.
21
Eigen vector corresponding to matrix A
�
1=
1
0
0
,�
2=
0
1
1
, �
3=
0
−1
1
Now to check whether �
1, �
2, �
3are pairwise orthogonal.
Pairwise Orthogonal –Two vectors �
1, �
2are said to be orthogonal
if their inner product i.e. �
1
??????
.�
2=0 
orthogonal pairwise are , , Here
also. vectorsothersfor check can weSimilarly
.orthogonal pairwise are , Therefore
0
1
1
0
001
1
1
0
0
0
1
.
321
21
21
XXX
XX
XX
T
T
=










=




















=

22. Nowweformnormalizedmodalmatrix
�=
10 0
0
1
2
−
1
2
0
1
2
1
2
,�
−1
=
10 0
0
1
2
1
2
0−
1
2
1
2
�
−1
��=
10 0
0
1
2
1
2
0−
1
2
1
2
100
03−1
0−13
10 0
0
1
2
−
1
2
0
1
2
1
2
22

23. 23
�
−1
��=
100
020
004
i.e. �
−1
��=D
where D is a diagonal matrix whose diagonal
elements are the eigen values of the matrix of
quadratic form.

24. The canonical Form
�
??????
(�
−1
��)�=�
??????
??????�
=�
1�
2�
3
100
020
004
�
1
�
2
�
3
=�
1
2
+2�
2
2
+4�
3
2
�
1
2
+2�
2
2
+4�
3
2
istherequiredcanonicalform.
24

25. Orthogonal
Transformation
�=��
�
1
�
2
�
3
=
�=��
�
1
�
2
�
3
=
10 0
0
1
2
−
1
2
0
1
2
1
2
�
1
�
2
�
3
�
1=�
1,
�
2=
1
2
�
2−
1
2
�
3,
�
3=
1
2
�
2+
1
2
�
3
istherequired
orthogonaltransformation.
25

26. Reduce 6�
2
+3�
2
+3�
2
−4��−2��+4��into
canonical form by orthogonal transformation.
Sol.Writethecoefficientmatrixofgivenquadratic
form
�=
6−22
−23−1
2−13
TheeigenvaluesofAare2,2,8
The eigen vectors corresponding to 2,2,8are
−1
0
2
,
1
2
0
and
2
−1
1
respectively.
26

27. 27
�
1=
−1
0
2
, �
2=
1
2
0
, �
3=
2
−1
1
Here �
1, �
2, are not pairwise orthogonal
Let �
1is a vector which is orthogonal to �
2and �
3
Let �
1=
�
�
�
than �
1
??????
.�
2= 0 or ���
1
2
0
=0
or a+2b=0……(1)
similarly�
1
??????
.�
3= 0 or ���
2
−1
1
=0 or 2a -b + c=0…..(2)
Assuming b =�,�������??????�������
�=−2�����=5�or �
1=
−2�
�
5�
or if �=1 than �
1=
−2
1
5
Now �
1, �
2, and �
3are pairwise orthogonal .
Now rest of the process is same as in previous example.

28. INDEX ,SIGNATURE,RANK
Let�=�
??????
��
beaquadraticforminnvariables�
1,�
2,...........�
??????.
INDEX∶Theno.ofpositiveterms(p)inthecanonicalform.
Signature:Thedifferencebetweenpositiveandnegativetermsinthecanonicalform.
Rank:A matrix is said to be of rank r when
(i) it has at least one non-zero minor of order r,
and (ii) every minor of order higher than r vanishes.
Briefly, the rank of a matrix is the largest order of any non-vanishing minor of
the matrix.
28

29. 29
Determine the rank of the following matrices:
(i)A=
���
���
���
Sol. (i)HereA is a 3 x3 matrix
∴??????(�)≤3
After operating �
2→�
2-�
1,�
3→�
3−2�
1
A~
���
��−�
��−�
Operating �
3→�
3-�
2
A~
���
��−�
���
The only third order minor is zero, but the second order minor
12
02
=2≠0
∴??????�=2

30. 30
Sol. (ii)Here B is a 2 x 4 matrix.
∴??????(�)≤2, the smaller of 2 and 4.
The second order minor
12
−20
=4≠0
∴??????�=2
(ii) B=
����
−����

31. Let �
??????
��be a real quadratic form in n
variables�
1,�
2,….�
??????with rank r and signature s.
Then we say that the quadratic form is
(i) positive definite if r = n, s = n
(ii) negative definite if r = n, s = 0
(iii) positive semidefinite if r<n and s = r
(iv) negative semidefinite if r<n, s = 0
(v) indefinite in all other cases.
31
NATURE OF QUADRATIC FORM
( using rank, signature)

32. NATURE OF QUADRATIC FORM
(using eigen values)
POSITIVE DEFINITE: If all the eigenvalues of
the coefficient matrix are POSITIVE.
POSITIVE SEMI DEFINITE: If all the eigen
values of the coefficient matrix are POSITIVE and
at least one is zero.
NEGATIVE DEFINITE: If all the eigenvalues
of the coefficient matrix are NEGATIVE.
NEGATIVE SEMI DEFINITE : If all the eigen
values of the coefficient matrix are NEGATIVE
and
at least one is zero.
➢INDEFINITE: If some of the eigenvalues are
POSITIVE and some are NEGATIVE.
32

33. EX.Determinethenature,indexandsignatureofthequadraticform
&#3627408452;(&#3627408485;)=2&#3627408485;
1&#3627408485;
2+2&#3627408485;
1&#3627408485;
3+2&#3627408485;
3&#3627408485;
2
Solution.Here
&#3627408436;=
011
101
110
NowthecharacteristicequationforAis
??????
3
−3??????−2=0
or ??????=2,−1,−1
Someoftheeigenvaluesarepositiveand
somearenegative.
Hence??????(??????)isindefinite.
Herethecanonicalformis2&#3627408485;
1
2
−&#3627408485;
2
2
−&#3627408485;
3
2
.
Index=1 No.ofpositivetermincanonicalform
Signature=-1
(differenceofpositiveandnegativetermincanonicalform)
33