Page 310 - Advanced engineering mathematics

290 CHAPTER 9 Eigenvalues, Diagonalization, and Special Matrices

THEOREM 9.12

1. The eigenvalues of a hermitian matrix are real.
2. The eigenvalues of a skew-hermitian are pure imaginary.

Proof By Lemma 9.1, an eigenvalue λ of any n × n matrix A, with corresponding eigenvector
E, satisfies

t
E AE
λ = t .
E E
We know that the denominator of this quotient is a positive number. Now use Lemma 9.2. If A
is hermitian, the numerator is real, so λ is real. If A is skew-hermitian then the numerator is pure
imaginary, so λ is pure imaginary.

Figure 9.2 shows a graphical representation of these conclusions about eigenvalues of matri-
ces. When plotted as points in the complex plane, eigenvalues of a unitary (or orthogonal)
matrix lie on the unit circle about the origin, eigenvalues of a hermitian matrix lie on the hor-
izontal (real) axis, and eigenvalues of a skew-hermitian matrix are on the vertical (imaginary)
axis.

9.3.4 Quadratic Forms

A quadratic form is an expression
n n

a jk z j z k
j=1 k=1

in which the a jk s and the z j s are complex numbers. If these quantities are all real, we say
that we have a real quadratic form.

Imaginary axis
Skew-hermitian
eigenvalues
i

1
Real axis
Hermitian
eigenvalues
Unitary
eigenvalues

FIGURE 9.2 Eigenvalues of unitary,
hermitian, and skew-hermitian matrices.

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October 14, 2010 14:49 THM/NEIL Page-290 27410_09_ch09_p267-294