0
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ It's true for all operator norms. $\endgroup$ – Christopher A. Wong Jun 8 '13 at 12:06
3
$\begingroup$

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 $|\lambda|\le\|A\|$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy