# Spectral radius and operator norm

Consider a FINITE endomorphism $A$ , then I was wondering whether the relation between the operator norm and the spectral radius $\rho$, given by: $\|A\| \ge \rho(A)$ is true for all operator norms or only the 2-norm?

• It's true for all operator norms. – Christopher A. Wong Jun 8 '13 at 12:06

It is true, not only for all operator norms, but also for all submultiplicative matrix norms: for any eigenpair $(\lambda,x)$ of $A$, repeat $x$ to form the columns of a square matrix $X$. Then $|\lambda|\|X\|=\|\lambda X\|=\|AX\|\le\|A\|\|X\|$ and hence $|\lambda|\le\|A\|$.