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

2025/07/03 by Tengjie Zhang, Zhang, Teng · 2 citations
Computer Science · Engineering · Mathematics · #15A60 #47A30 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Matrix Theory and Algorithms #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2507.02684

openalex publication_date 2025/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 2010, Eun-Young Lee conjectured that if A,B are two n× n complex matrices and |A|, |B| are the absolute values of A, B, respectively, then ‖A+B‖_F≤ √\dfrac1+√(2)2‖|A|+|B|‖_F, where ‖⋅‖_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.

Cited by

• Generalizing Lee's conjecture on the sum of absolute values of matrices
• Sharp perturbation bounds on the Frobenius norm of subunitary and positive polar factor

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