# Banach's Lemma

1. Nov 18, 2013

### twoski

1. The problem statement, all variables and given/known data

This isn't exactly homework per se, i just am having a hard time figuring out how to solve these types of questions generally...

Consider the 3x3 matrix A = diag[1,3,1]. Show that this matrix is invertible no matter what its dimensions are. Determine an upper bound on $||A^{-1}||_{∞}$ and on the condition number of A

3. The attempt at a solution

So from what i've picked up on so far, i start out:

A = D + E

where D contains all values of A that are in the same row and column (ie. 1,1... 2,2... etc) and E contains all values of A that are not on the diagonal.

Next, say that B = $D^{-1}$ * E and A = D(I + B)

It follows that $||B||_{∞}$ < 1 because of diagonal dominance. Hence A is invertible (I have no idea what diagonal dominance means in this context, i just wrote down what i'm supposed to say in order to get marks here - not much explanation has been given on this part).

How do i find the condition number and upper bounds?

2. Nov 20, 2013

### twoski

So apparently the condition number is determined by adding the infinity norm of A with the infinity norm of its inverse - is this correct?