¦

¦

Generally, it is clear that any Hermitian matrix is such that is real for any vector

B

C

£

‘

¦

. It turns out that the converse is also true, i.e., it can be shown that if is

C

t

‘

real for all vectors in , then the matrix is Hermitian; see Exercise 15.

Eigenvalues of Hermitian matrices can be characterized by optimality properties of

the Rayleigh quotients (1.28). The best known of these is the min-max principle. We now

label all the eigenvalues of in descending order:

(6 ( ‘ (

0@ @ @

§§¥

¥¥

Here, the eigenvalues are not necessarily distinct and they are repeated, each according to

its multiplicity. In the following theorem, known as the Min-Max Theorem, represents a

‘

generic subspace of .

z„ §CjG b¤

¥¦ ¥¡ Q U

T

The eigenvalues of a Hermitian matrix are characterized by the

relation

C gx B B ( §¦¥¢ ©¢¤

¢ ¢

©¦

¦

t

¢ ' ¢ ' ¨ £ ¢ §¥

¦

(

@ & $% "

(" 4

( 0' ¦© ¦

&‘ C

nd¥ ¥ ”˜

uu d

u £

¡£ ¥ ¢ ¡ ¥ ¥ ¥ § ¥ ¤© ¡

¡£¥ ¡

¥

§

©

¥

£ 1( % !

0

©¨¦

§ §¥ T

‘ '

" %%$%"

$$

)

Let be an orthonormal basis of consisting of eigenvectors of

‘ 111) 0@

associated with respectively. Let be the subspace spanned by the ¬rst of @

B ‘

gB C x B

© ¦" (

¦© ¦ ¦

these vectors and denote by the maximum of over all nonzero vectors ¦

C

C '

of a subspace . Since the dimension of is , a well known theorem of linear algebra

—˜ ˜ ) ' ) ’ 3

shows that its intersection with any subspace of dimension is not reduced to

01(

!7

' ¦¡

¦ ¦

, i.e., there is vector in . For this , we have

6 ')

( 6 0 1(0 ' ( ) C xx B B

©¦

¦

@

' @

©

¦¦

C

(

B '@

so that .

¦

– ˜

C —3

Consider, on the other hand, the particular subspace of dimension which

&

1)11 '

) ) ¦

is spanned by . For each vector in this subspace, we have

‘ 6 §( ‘ ) ) C g B 6

¦

¦ ' @ '@

g B

'§( ‘ #

¦¦

C C ’ ˜ B

—

B 3

'@

so that . In other words, as runs over all the -dimensional

& #

(

¦

C

B B

'@

subspaces, is always and there is at least one subspace for which

& & #

¦ ¦

C C

'@. This shows the desired result.

The above result is often called the Courant-Fisher min-max principle or theorem. As a

particular case, the largest eigenvalue of satis¬es

C gx B B §¦¥¢ 0

¤ ¦© ¦

t

©¦ ¢ §¥

¨¦

(

@ '$

& ¦© ¦

C

Actually, there are four different ways of rewriting the above characterization. The

second formulation is

g B

¢ ©¢ §( ¢ ¤¦ ¢ ¢ ' ¢ ' ¦

¦