Some generalizations of the DDVV and BW inequalities

TL;DR

This paper extends the DDVV and BW inequalities to broader classes of matrices, including quaternionic matrices and those related to Clifford algebras, providing new bounds and generalizations.

Contribution

It introduces generalized DDVV and BW inequalities applicable to real, complex, and quaternionic matrices, expanding their scope and applicability.

Findings

Generalized DDVV inequalities for matrices in Clifford algebra subspaces

Extended Böttcher-Wenzel inequality to quaternionic matrices

New bounds for matrix commutators in various algebraic settings

Abstract

In this paper we generalize the known DDVV-type inequalities for real (skew-)symmetric and complex (skew-)Hermitian matrices to arbitrary real, complex and quaternionic matrices. Inspired by the Erd\H{o}s-Mordell inequality, we establish the DDVV-type inequalities for matrices in the subspaces spanned by a Clifford system or a Clifford algebra. We also generalize the B\"{o}ttcher-Wenzel inequality to quaternionic matrices.