B ¨

&

2

¦

¦ ¦© ¦

C

‘

de¬ned for any nonzero vector in . These ratios are known as Rayleigh quotients and

are important both for theoretical and practical purposes. The set of all possible Rayleigh

t

‘

¦

quotients as runs over is called the ¬eld of values of . This set is clearly bounded

C B 6 # B

‘

¦ ¦ ¦

since each is bounded by the the 2-norm of , i.e., for all .

¦ ¦

C

¦

If a matrix is normal, then any vector in can be expressed as

‘& )

1(

0

B

) "

¦

where the vectors form an orthogonal basis of eigenvectors, and the expression for ¦

C

becomes

‘ ) ) C x B

6 ' '

‘&

©¦

¦

t §¦¥

' ' 0 )(' 1)'

0(

6 ' @ ¦

B ¨

¢&

¦

g B @

¦

‘

¦¦ 1)'

0(

C Y

C

where

‘ & ¤ Y # 6 ' 6 1( ' )

‘ 1( ()

¦ ¦

#7 (

0‘

0

From a well known characterization of convex hulls established by Hausdorff (Hausdorff™s

¦

convex hull theorem), this means that the set of all possible Rayleigh quotients as runs

t

‘

over all of is equal to the convex hull of the ™s. This leads to the following theorem @

which is stated without proof.

SG b¡¤

§¥¦ ¥Q T

The ¬eld of values of a normal matrix is equal to the convex hull of its

spectrum.

The next question is whether or not this is also true for nonnormal matrices and the

answer is no: The convex hull of the eigenvalues and the ¬eld of values of a nonnormal

matrix are different in general. As a generic example, one can take any nonsymmetric real

matrix which has real eigenvalues only. In this case, the convex hull of the spectrum is

a real interval but its ¬eld of values will contain imaginary values. See Exercise 12 for

another example. It has been shown (Hausdorff) that the ¬eld of values of a matrix is a

convex set. Since the eigenvalues are members of the ¬eld of values, their convex hull is

contained in the ¬eld of values. This is summarized in the following proposition.

uY u d¥ £ ¡ ¨ ¢¤¥

n ¦¢¥

¤¤§ ¥ §¢

¡

£ ¡© ¥ ¡ © ¥ ¡ ¥ © ¥ ¥

¥ ¨

q¤ G S£

£¦

£ U SRFP cbP` Q`

HQ F Q

T

The ¬eld of values of an arbitrary matrix is a convex set which

contains the convex hull of its spectrum. It is equal to the convex hull of the spectrum

when the matrix is normal.

¦¨

B¥§¢

A¡ ¦¡ P0§ ' 6 G' G 6¥¨%

§

A ¬rst result on Hermitian matrices is the following.

§CjG b¤

¥¦ ¥¡

Q ¡U

T

¡ C B

The eigenvalues of a Hermitian matrix are real, i.e., .

X

§ § ¥¦£

T

Let be an eigenvalue of and an associated eigenvector or 2-norm unity.

@ V

Then

C B C B C %

B

@ W

V V V W

V W

V V @

which is the stated result.

It is not dif¬cult to see that if, in addition, the matrix is real, then the eigenvectors can be

chosen to be real; see Exercise 21. Since a Hermitian matrix is normal, the following is a

consequence of Theorem 1.7.

„ §CjG b¤

¥¦ ¥¡ ¡U

T

Q

Any Hermitian matrix is unitarily similar to a real diagonal matrix.

In particular a Hermitian matrix admits a set of orthonormal eigenvectors that form a basis

j

‘

of .

B

In the proof of Theorem 1.8 we used the fact that the inner products are real. W

V V

g B