1. Show that if T is the Cholesky decomposition of the positive definite matrix then -1T −1 is symmetric. 2. 3. The diagonal preconditioner, also called the Jacobi preconditioner, is one of the...

1.

Show that if

^T

is the Cholesky decomposition of the positive definite matrix

then

<sup>-1

T

−1</sup>
is symmetric.

2.

3.

The diagonal preconditioner, also called the Jacobi preconditioner, is one of the simplest means for reducing the condition number of the coefficient matrix. The method is particularly effective for a diagonally dominant matrix or a matrix with widely different diagonal elements but, of course, the method does not always help.

Let

be the diagonal of

and use

⁻¹
as the preconditioner. Develop a formula for the method that does not involve computing

⁻¹

What happens if all elements of the diagonal are constant