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arXiv · 2507.02684

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

Teng Zhang
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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.

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Teng Zhang. 2025-07-03. A new proof of Lee's conjecture on the Frobenius norm via the matrix Cauchy-Schwarz inequality. https://arxiv.org/abs/2507.02684

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