Pseudospectra and Simultaneous Power Control

TL;DR

This paper demonstrates that matrices with identical pseudospectra can exhibit significantly different behaviors when raised to powers, impacting power control strategies in matrix analysis.

Contribution

It introduces the construction of matrix pairs with identical pseudospectra but divergent power norms, extending to more general functions under certain conditions.

Findings

Existence of 10x10 matrices with identical pseudospectra and divergent power norms

Divergence persists for multiple powers, e.g., n and m

Results extend to more general functions under specific conditions

Abstract

We prove that, for $M>0$ and $n,m\ge2$, we can find two $10\times10$ matrices $A$ and $B$ with identical pseudospectra such that we have simultaneously $\|A^n\|/\|B^n\|>M$ and $\|A^m\|/\|B^m\|>M$. We also prove that, under certain conditions, this result holds for two more general functions of $A$ and $B$.