A new proof of Lee's conjecture on the Frobenius norm via the matrix Cauchy-Schwarz inequality

TL;DR

This paper offers a new proof of Lee's conjecture on the Frobenius norm inequality for matrices, using only the Cauchy-Schwarz inequality, simplifying the previous proof that involved angle-based inequalities.

Contribution

It provides a novel proof of Lee's conjecture relying solely on the Cauchy-Schwarz inequality, avoiding angle-based arguments used previously.

Findings

Confirmed Lee's conjecture with a new proof

Simplified the proof technique using basic inequalities

Strengthened understanding of Frobenius norm inequalities

Abstract

In 2010, Eun-Young Lee conjectured that if $A,B$ are two $n\times n$ complex matrices and $\left|A\right|, \left|B\right|$ are the absolute values of $A, B$, respectively, then \[ \|A+B\|_F\le \sqrt{\dfrac{1+\sqrt{2}}{2}}\|\left|A\right|+\left|B\right|\|_F, \] where $\|\cdot\|_F$ is the Frobenius norm of matrices. This conjecture has been proven by Lin and Zhang [J. Math. Anal. Appl. 516 (2022) 126542] by studying inequalities for the angle between two matrices induced by the Frobenius inner product. In this paper, we present a new proof of the same result, relying solely on the Cauchy-Schwarz inequality.