Page 308 - Advanced engineering mathematics

288 CHAPTER 9 Eigenvalues, Diagonalization, and Special Matrices

This means that the eigenvalues of U lie on the unit circle about the origin in the complex
plane. Since a real orthogonal matrix is also unitary, this also holds for real orthogonal matrices.
Proof Let λ be an eigenvalue of U with eigenvector E. We know that UE=λE. Then UE=λE.
Therefore,
t
t
(UE) = λ(E) .
Then,
t
t
t
(E) (U) = λ(E) .
t
−1
But U is unitary, so U = U . The last equation becomes
t
t
−1
(E) U = λ(E) .
Multiply both sides of this equation on the right by UE:
t
t
t
t
−1
(E) U UE = λ(E) UE = λ(E) λE = λλE E.
t
Now E E is the dot product of an eigenvector with itself, and so is a positive number. Dividing
t
the last equation by E E yields the conclusion that λλ=1. Then |λ| =1, proving the theorem.
2
9.3.3 Hermitian and Skew-Hermitian Matrices
t
An n × n complex matrix H is hermitian if H = H .
That is, a matrix is hermitian if its conjugate equals its transpose. If a hermitian matrix has
real elements, then it must be symmetric, because then the matrix equals its conjugate, which
equals its transpose.

t
An n × n complex matrix S is skew-hermitian if S =−S .

Thus, S is skew-hermitian if its conjugate equals the negative of its transpose.

EXAMPLE 9.15
The matrix
⎛ ⎞
15 8i 6 − 2i
H = ⎝ −8i 0 −4 + i ⎠
6 + 2i −4 − i −3
is hermitian because
⎛ ⎞
15 −8i 6 + 2i
t
H = ⎝ 8i 0 −4 − i ⎠ = H .
6 − 2i −4 + i −3
The matrix
⎛ ⎞
0 8i 2i
S = 8i 0 4i ⎠
⎝
2i 4i 0

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October 14, 2010 14:49 THM/NEIL Page-288 27410_09_ch09_p267-294