A new proof of some matrix inequalities

TL;DR

This paper provides alternative proofs for certain matrix inequalities involving spectral properties and determinants, under specific conditions, enhancing understanding of matrix spectral bounds.

Contribution

It introduces new proofs for well-known matrix inequalities, expanding the theoretical framework and offering alternative approaches to spectral inequality proofs.

Findings

Established bounds on eigenvalues of matrix products.

Derived integral inequalities involving determinants and spectra.

Provided conditions under which the inequalities hold.

Abstract

In this paper we give alternate proofs of some well-known matrix inequalities. In particular, we show that under certain conditions the inequality holds \begin{align}\sum \limits_{\lambda_i\in \mathrm{Spec}(ab^{T})}\mathrm{min}\{\log |t-\lambda_i|\}_{[||a||,||b||]}&\leq \# \mathrm{Spec}(ab^T)\log\bigg(\frac{||b||+||a||}{2}\bigg)\nonumber \\&+\frac{1}{||b||-||a||}\sum \limits_{\lambda_i\in \mathrm{Spec}(ab^T)}\log \bigg(1-\frac{2\lambda_i}{||b||+||a||}\bigg).\nonumber \end{align}Also under the same condition, the inequality also holds\begin{align}\int \limits_{||a||}^{||b||}\log|\mathrm{det}(ab^{T}-tI)|dt&\leq \# \mathrm{Spec}(ab^T)(||b||-||a||)\log\bigg(\frac{||b||+||a||}{2}\bigg)\nonumber \\&+\sum \limits_{\lambda_i\in \mathrm{Spec}(ab^T)}\log \bigg(1-\frac{2\lambda_i}{||b||+||a||}\bigg).\nonumber \end{align}