On a determinantal inequality arising from diffusion tensor imaging

TL;DR

This paper proves a new determinantal inequality involving positive semidefinite matrices, complementing a previous inequality, and explores the relationship between determinantal inequalities and majorization relations.

Contribution

It introduces a novel inequality relating determinants of matrices with absolute values, expanding the theoretical understanding of matrix inequalities in diffusion tensor imaging.

Findings

Proves a new determinantal inequality for positive semidefinite matrices.

Establishes the interplay between determinantal inequalities and majorization.

Provides insights into related open questions in matrix analysis.

Abstract

In comparing geodesics induced by different metrics, Audenaert formulated the following determinantal inequality $$\det(A^2+|BA|)\le \det(A^2+AB),$$ where $A, B$ are $n\times n$ positive semidefinite matrices. We complement his result by proving $$\det(A^2+|AB|)\ge \det(A^2+AB).$$ Our proofs feature the fruitful interplay between determinantal inequalities and majorization relations. Some related questions are mentioned.