A Generalization of the B\"{o}ttcher-Wenzel inequality for three rectangular matrices

TL;DR

This paper extends the Böttcher-Wenzel inequality to three rectangular matrices, providing bounds on the Frobenius norm of their generalized commutator, with tighter bounds in special cases.

Contribution

It introduces a generalized inequality for the Frobenius norm of the commutator of three matrices, broadening the scope of the original Böttcher-Wenzel inequality.

Findings

Derived a new inequality for the Frobenius norm of $ABC - CBA$

Provided tighter bounds when one matrix dimension is 1

Generalized the Böttcher-Wenzel inequality to three matrices

Abstract

Let $m,n$ be positive integers. For all $m\times n$ complex matrices $A, C$ and an $n\times m$ matrix $B$, we define a generalized commutator as $ABC-CBA$. We estimate the Frobenius norm of it, and finally get the inequality, which is a generalization of the B\"{o}ttcher-Wenzel inequality. If $n=1$ or $m=1$, then the Frobenius norm of $ABC-CBA$ can be estimated with a tighter upper bound.