Sharp perturbation bounds on the Frobenius norm of subunitary and positive polar factor

TL;DR

This paper establishes precise perturbation bounds for the Frobenius norm of subunitary and positive polar factors, improving classical inequalities and conjectures using convex analysis and singular value insights.

Contribution

It provides sharp upper and lower bounds for perturbations of polar factors, refining existing inequalities and conjectures with a novel analytical approach.

Findings

Derived sharp bounds for Frobenius norm perturbations.

Refined classical inequalities and strengthened conjectures.

Generalized matrix inequalities into tighter forms.

Abstract

Leveraging tools from convex analysis and incorporating additional singular value information of matrices, we completely resolve the problem of establishing perturbation bounds for the Frobenius norm of subunitary and positive polar factors. We derive corresponding sharp upper and lower bounds. As corollaries, we refine the results of Li and Sun [SIAM J. Matrix Anal. Appl., 23 (2002), pp. 1183--1193] and strengthen the classical Araki-Yamagami inequality [Comm. Math. Phys., 81 (1981), no. 1, pp. 89--96]. The versatility of our method also allows us to strengthen Lee's conjecture, providing a sharper version along with a matching sharp lower bound. Furthermore, we generalize the classical matrix arithmetic-geometric mean inequality and Cauchy-Schwarz inequality into tighter and more robust forms. Finally, we establish a sharp lower bound for a result by Kittaneh [Comm. Math. Phys., 104 (1986), no. 2, pp. 307--310].