Looks fine to me!
(By the way, Wikipedia has some thoughts on this problem: Spectral radius - Wikipedia, the free encyclopedia.)
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
.
Looks fine to me!
(By the way, Wikipedia has some thoughts on this problem: Spectral radius - Wikipedia, the free encyclopedia.)