I have been trying to understand matrix norms for a while and I do not find it easy.

If someone has a good writeup on the subject for us slow ones, please share.

Anyway, here's the problem I'm currently working on:

problem:

Let denote any norm on and also the induced matrix norm on .

Show that , where is the spectral radius of , i.e., the largest absolute value of an eigenvalue of .

attempt:

- I take the norm on both sides. (I don't really know if this is a valid step)

I now use the fact that is the smallest number for which the inequality (1) holds for any

(1)

and so

.