15.4. THE RAYLEIGH QUOTIENT 357

To see if more iteration would be needed, check this. 2 1 32 1 13 2 1

 1.0

0.654120.95404

=

 5.51623.60825.2623

and

5.5162

 1.00.654120.95404

=

 5.51623.60835.2627

Thus this is essentially an eigenvector with eigenvalue equal to 5.5162.

15.4 The Rayleigh QuotientThere are many specialized results concerning the eigenvalues and eigenvectors for Hermi-tian matrices. A matrix A is Hermitian if A = A∗ where A∗ means to take the transpose ofthe conjugate of A. In the case of a real matrix, Hermitian reduces to symmetric. Recallalso that for x ∈ Fn,

|x|2 = x∗x =n

∑j=1

∣∣x j∣∣2 .

The following corollary gives the theoretical foundation for the spectral theory of Her-mitian matrices. This is a corollary of a theorem which is proved Corollary 13.2.14 andTheorem 13.2.14 on Page 308.

Corollary 15.4.1 If A is Hermitian, then all the eigenvalues of A are real and there existsan orthonormal basis of eigenvectors.

Thus for {xk}nk=1 this orthonormal basis,

x∗i x j = δ i j ≡

{1 if i = j0 if i ̸= j

For x ∈ Fn, x ̸= 0, the Rayleigh quotient is defined by

x∗Ax|x|2

.

Now let the eigenvalues of A be λ 1 ≤ λ 2 ≤ ·· · ≤ λ n and Axk = λ kxk where {xk}nk=1 is

the above orthonormal basis of eigenvectors mentioned in the corollary. Then if x is anarbitrary vector, there exist constants, ai such that

x =n

∑i=1

aixi.

Also,

|x|2 =n

∑i=1

aix∗in

∑j=1

a jx j = ∑i j

aia jx∗i x j = ∑i j

aia jδ i j =n

∑i=1|ai|2 .