Next: The QR Method Up: The Rayleigh-Ritz Method: Previous: The Rayleigh-Ritz Method:

### Rayleigh-Ritz, Background:

Let's consider more generally the case for an Hermitian matrix and is an dimensional vector. We indicate hermitian'' as , which means that the complex conjugate of the transpose of is the same as . So, for short, . If is a real matrix, . These matrices often arise from self-adjoint continuous operators which model some physical process. The complex version appears often in the context of quantum mechanics and acoustics.

We will indicate by an overbar the operation of taking the conjugate transpose. If and were real, this operation would simply involve the transpose.

Since the eigenvalues are real and can be organized as

We will see that the smallest and largest eigenvalues may be thought of as the solution to a constrained minimum and maximum problem.

Theorem (Rayleigh-Ritz): Let as above and the eigenvalues ordered as above. Then

Furthermore,

and

Proof: Since then there exists a unitary matrix such that , with . For any we have

Since is non-negative, then

Because is unitary

Hence,

These are sharp inequalities. If is an eigenvector of associated with , then

Same sort of argument holds for .

Furthermore, if then

so
 (85)

Finally, since , then

and

Hence, (86) is equivalent to

Same sort of arguments hold for , in the context of the minimum.

Algorithm

Now we will revert to the case of an symmetric real matrix for the presentation of the algorithm.

Let be an dimensional real vector. Choose some initial guess , and compute

then

where is the inner product.

In fact, by writing then

hence, it is easy to see that

which is quadratic convergence, an improvement over the previous method.

Next: The QR Method Up: The Rayleigh-Ritz Method: Previous: The Rayleigh-Ritz Method:
Juan Restrepo 2003-04-12