Take a look at Kreyszig's Introductory Functional Analysis with Applications, pages 307 to 312. That's one approach (Banach fixed-point theorem).
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.
Well, I was speaking to some other students and they said that, since it just says matrix norm I can use the max norm. This makes the proof quite easy.
Since the matrix is strictly diagonally dominant we have
Thus the Jacobi method is convergent. Gauss-Seidel is similar.