Remarks on the Bottcher-Wenzel Inequality

TL;DR

This paper presents a new, highly conceptual proof of the Bottcher-Wenzel inequality concerning the Frobenius norm of commutators of real matrices, and discusses related inequalities like the Chern-do Camo-Kobayashi inequality.

Contribution

It provides a novel, minimal-computation proof of the Bottcher-Wenzel inequality and explores related matrix norm inequalities.

Findings

New proof of the Bottcher-Wenzel inequality

Discussion of related inequalities like Chern-do Camo-Kobayashi

Minimal computational approach to matrix norm inequalities

Abstract

In 2005, B\"ottcher and Wenzel raised the conjecture that if $X,Y$ are real square matrices, then $||XY-YX||^2\leq 2||X||^2||Y||^2$, where $||\cdot||$ is the Frobenius norm. Various proofs of this conjecture were found in the last few years by several authors. We here give another proof. This proof is highly conceptual and requires minimal computation. We also briefly discuss related inequalities, in particular, the classical Chern-do Camo-Kobayashi inequality.