Harnack type inequalities for matrices in majorization

TL;DR

This paper extends and refines Harnack type inequalities for matrices using majorization techniques, providing stronger bounds and exploring related inequalities with applications to spectral norms and Cayley transforms.

Contribution

It offers new, more general bounds on Harnack inequalities for matrices based on eigenvalue and singular value majorization, with simplified proofs and additional related inequalities.

Findings

Derived stronger bounds on Harnack inequalities for matrices.

Compared different methods for obtaining bounds and discussed their implications.

Proposed open problems related to spectral norm and eigenvalues.

Abstract

Following the recent work of Jiang and Lin (Linear Algebra Appl. 585 (2020) 45--49), we present more results (bounds) on Harnack type inequalities for matrices in terms of majorization (i.e., in partial products) of eigenvalues and singular values. We discuss and compare the bounds derived through different ways. Jiang and Lin's results imply Tung's version of Harnack's inequality (Proc. Amer. Math. Soc. 15 (1964) 375--381); our results %with simpler proofs are stronger and more general than Jiang and Lin's. We also show some majorization inequalities concerning Cayley transforms. Some open problems on spectral norm and eigenvalues are proposed.