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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
Real and Complex Matrices 3-3
Theorem 3.8 The determinant of an orthogonal
matrix has the value +1 or 1.
Proof We know that
det AB = det A · det B and
det A
T
= det A (3.11), (3.12)
If A is an orthogonal matrix then A
T
· A = I
1 = det I = det A
T
A = det A
T
· det A = (det A)
2
det A = ±1.
Theorem 3.9 The eigenvalues of an orthogonal
matrix are of unit modulus.
Proof Let A be an orthogonal matrix A
T
A = I
(3.13)
If l is an eigenvalue and X 0 is the eigenvector
corresponding to l then
AX = lX (3.14)
Taking transposes of both sides of (3.14)
(AX)
T
= (lX)
T
= lX
T
X
T
A
T
= lX
T
(3.15)
Multiplying the LH and RH members of equations
(3.15) and (3.14), ...
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Publisher Resources

ISBN: 9781299446557Publisher Website