# Math Help - Show that rho(A) <= ||A||

1. ## Show that rho(A) <= ||A||

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 $\mathbb{C}^m$ and also the induced matrix norm on $\mathbb{C}^{m\times m}$.
Show that $\rho(A)\leq ||A||$, where $\rho(A)$ is the spectral radius of $A$, i.e., the largest absolute value $|\lambda|$ of an eigenvalue of $A$.

attempt:
$Ax=\lambda x$
- I take the norm on both sides. (I don't really know if this is a valid step)
$||Ax||=||\lambda x||=|\lambda| ||x||$

I now use the fact that $||A||$ is the smallest number for which the inequality (1) holds for any $x\in\mathbb{C}^n$

(1) $||Ax||\leq C||x||$

$||Ax||=||\lambda x||=|\lambda| ||x||\leq C||x||$

and so
$
|\lambda|\leq C$
.

2. Looks fine to me!

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