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

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 $\Vert \cdot {\Vert }_F$ 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.

arXiv comment4 pages. This is the final version to appear in the Linear Algebra Appl