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

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

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

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

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

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

2. Looks fine to me!

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