Page 309 - Advanced engineering mathematics

9.3 Some Special Types of Matrices 289

is skew-hermitian because
⎛ ⎞
0 −8i −2i
t
S = −8i 0 −4i ⎠ =−S .
⎝
−2i −4i 0
We want to derive a result about eigenvalues of hermitian and skew-hermitian matrices.
For this we need the following conclusions about the numerator of the general expression for
eigenvalues in Lemma 9.1.

LEMMA 9.2

Let
⎛ ⎞
z 1
z 2
⎜ ⎟
⎜ ⎟
Z = ⎜ . ⎟
.
⎝ . ⎠
z n
be a complex n × 1 matrix. Then
t
1. If H is n × n hermitian, then Z HZ is real.
t
2. If S is n × n skew-hermitian, then Z HZ is pure imaginary.
t
Proof of Lemma 9.3 For condition (1), suppose H is hermitian, so that H = H. Then
t
t
t
(Z HZ) = ((Z) )HZ = Z HZ.
t
But Z HZ is a 1 × 1 matrix and so equals its own transpose. Continuing from the last equation,
we have
t
t
t
t
t
t
t
Z HZ = (Z HZ) = Z H (Z) = Z HZ.
This shows that
t t
(Z HZ) = Z HZ.
t
Since Z HZ equals its own conjugate, this quantity is real.
t
To prove condition (2), suppose S is skew-hermitian, so S =−S. By an argument like that
in the proof of condition (1), we find that
t t
(Z SZ) =−Z SZ
t
If we write Z SZ = a + ib, then the last equation means that
a − ib =−a − ib.
t
But then a =−a so a = 0 and Z SZ is pure imaginary. This includes the possibility of a zero
eigenvalue.

This lemma absorbs most of the work we need for the following result, giving us information
about eigenvalues.

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October 14, 2010 14:49 THM/NEIL Page-289 27410_09_ch09_p267-294