利用矩阵Cauchy-Schwarz不等式证明Lee关于Frobenius范数的猜想

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2025-07-18 DOI:10.1016/j.laa.2025.07.017

Teng Zhang

引用次数: 0

摘要

2010年，Eun-Young Lee推测若A、B是两个n×n复矩阵，且|A|、|B|分别是A、B的绝对值，则‖A+B‖F≤1+22‖|A|+|B|‖F，其中‖⋅‖F为矩阵的Frobenius范数。Lin和Zhang(2022)[3]通过研究由Frobenius内积引起的两个矩阵夹角的不等式证明了这一猜想。在本文中，我们仅依靠Cauchy-Schwarz不等式给出了对相同结果的一个新的证明。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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

In 2010, Eun-Young Lee conjectured that if

A, B

are two

n \times n

complex matrices and

| A |, | B |

are the absolute values of

A, B

, respectively, then

{‖ A + B ‖}_{F} \leq \sqrt{\frac{1 + \sqrt{2}}{2}} {‖ | A | + | B | ‖}_{F},

where

{‖ \cdot ‖}_{F}

is the Frobenius norm of matrices. This conjecture has been proven by Lin and Zhang (2022) [3] 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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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.