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Mathematical Methods by Pearson
book

Mathematical Methods by Pearson

by E. Rukmangadachari
May 2024
Intermediate to advanced content levelIntermediate to advanced
437 pages
17h 41m
English
Pearson India
Content preview from Mathematical Methods by Pearson
2-16 Mathematical Methods
Such a matrix is called an orthogonal matrix. It
satisfies P
T
P = I or P
1
= P
T
. The diagonalisation
of a square matrix A by an orthogonal matrix P is
called orthogonalisation of A.
Example 2.12
Find a modal matrix (orthogonal matrix) P and
diagonalise the symmetric matrix
11
11
A
⎡⎤
=
⎢⎥
⎣⎦
.
Solution The characteristic equation of A is
2
12
11
() (1 ) 1
11
( 2) 0 0, 2
P
l
ll
l
ll l l
==−−
=−=⇒= =
l
1
= 0
1
12
2
12
11 1
11 0
0
11 0
1
;
1
11
x
xx
x
xx
cX c
⎡⎤⎡⎤⎡⎤
=+=
⎢⎥⎢⎥⎢⎥
⎣⎦⎣⎦⎣⎦
⎡⎤
⇒== =
⎢⎥
⎣⎦
l
2
= 2
1
12
2
12
22 2
11 0
0
11 0
1
;
1
11
x
xx
x
xx
cX c
⎡⎤⎡⎤⎡⎤
=+=
⎢⎥⎢⎥⎢⎥
⎣⎦⎣⎦⎣⎦
⎡⎤
⇒== =
⎢⎥
⎣⎦
The eigenvectors
1
1
1
X
⎡⎤
=
⎢⎥
⎣⎦
and
2
1
1
X
⎡⎤
=
⎢⎥
⎣⎦
(c
1
= c
2
= 1)
are orthogonal.
12
1
.[
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Publisher Resources

ISBN: 9781299446557Publisher Website