©¨¦

§ §¥ T

The ¬rst statement is an immediate consequence of the de¬nition of positive de¬-

C B aW

7

niteness. Indeed, if were singular, then there would be a nonzero vector such that V

7

and as a result for this vector, which would contradict (1.41). We now prove

V V

the second part of the theorem. From (1.45) and the fact that is positive de¬nite, we

conclude that is HPD. Hence, from (1.33) based on the min-max theorem, we get

‘ ” ( C B §B ( % ¢ C B B ( % ¢ z

§ B

V V V V 7

( (

@

'$

& '$

&

C

V V V V

C

C

‘ ” §B

Taking yields the desired inequality (1.46).

@

C

Y

A simple yet important result which locates the eigenvalues of in terms of the spectra

uY u d¥ £ ¡ ¨ ¢¤¥

n ¦¢¥

© ¤¤§ ¥ §¢

¡

£ ¡© ¥ ¡ © ¥ ¡ ¥ © ¥ ¥

¥ ¨

of and can now be proved.

I B 0 6 §C jG b¤

¥¦ ¥¡ s B 06

—

Q U

T

Let be any square (possibly complex) matrix and let

…

3

and . Then any eigenvalue of is such that

I @

C

C

¨ £ t §¥

‘ ” # … B7 ¢

§ B ¦ B ¦

@¤ (

#

@ @

C C B ¡”” # C C … B ¤ ¨ £ t §¥

# B

C ¦ 2

@ @ @

C ‘” ¡

£

§ § ¥¦ T

When the decomposition (1.42) is applied to the Rayleigh quotient of the eigen-

… …

vector associated with , we obtain

V @

C … … )B — C … … §B C … … B … t

¨ ¢ £ §¥

¦

@ W

V V V V c

V V

6 …

assuming that . This leads to

V

…C … §B C … B7 ¢ @¤ V V

C … ‘… B C … B ¤ @ c

V V

The result follows using properties established in Section 1.9.

Thus, the eigenvalues of a matrix are contained in a rectangle de¬ned by the eigenval-

ues of its Hermitian part and its non-Hermitian part. In the particular case where is real,

then is skew-Hermitian and its eigenvalues form a set that is symmetric with respect to

the real axis in the complex plane. Indeed, in this case, is real and its eigenvalues come

in conjugate pairs.

Note that all the arguments herein are based on the ¬eld of values and, therefore,

they provide ways to localize the eigenvalues of from knowledge of the ¬eld of values.

However, this approximation can be inaccurate in some cases.

¥

© U © §¦

T ¥

Consider the matrix

£¢ 7

C — ¢ •B

¥¥ (

3 7

¨

The eigenvalues of are and 101. Those of are and those of

• (

(

¢¤7 )

B 3

¨

are .

C

˜

When a matrix is Symmetric Positive De¬nite, the mapping

t

x

xB

g ˜B

¤ ¨ £ ¤ §¥

¦

¨© ¦ ¨© ¦ ¦

¨

£

C C

Y

jlj

‘ ‘ ‘