I am studying for my last Qualifying exam and came across two related problems that I can't seem to solve, both are Iterative Methods for Linear Systems.

Given A is a strictly diagonally dominant matrix, then show that the Gauss Seidel and Jacobi methods converge.

Jacobi:

Let where

Thus,

and the iterative method is given by

.

Gauss-Seidel:

Let where

Thus,

and the iterative method is given by

.

I have proven that if we have an iterative method

converges if where is a matrix norm. Thus, all I would have to do is prove that and .

However, I have a theorem that says the following are equivalent:

1. converges

2.

So if I can prove that the spectral radius is less than one I have shown that the method converges. I don't really care which way I prove this, but I can't figure out how to do it.