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

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.