1. Prove Lemma 19.3. Hint: Look at it as a problem involving block matrices. 2. Let be a positive definite matrix and T be the Cholesky decomposition. Using the singular value decomposition of show...

1.

Prove Lemma 19.3. Hint: Look at it as a problem involving block matrices.

2.

Let

be a positive definite matrix and

^T

be the Cholesky decomposition. Using the singular value decomposition of

show how to compute the eigenvalues and associated eigenvectors of

3.

If
is positive definite, the Cholesky decomposition can be used to determine the eigenvalues of

Here is an outline of the algorithm:

It can be shown that the sequence converges to a diagonal matrix of eigenvalues.

Prove that if

is positive definite, then any matrix similar to

is also positive definite.

What can you say about the eigenvalues of a positive definite matrix

Show that each matrix,

_i

is similar to

so that upon termination of the algorithm,

and Amaxiter have the same eigenvalues.