关于加权Frobenius范数Böttcher-Wenzel不等式的若干猜想

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2025-03-24 DOI:10.1016/j.laa.2025.03.015

Wenbo Fang, Che-Man Cheng

引用次数: 0

摘要

设ω是一个正定矩阵。ω加权Frobenius范数‖⋅‖ω定义为‖X‖ω= ωX X。最近，A. Mayumi， G. Kimura， H. Ohno和D. Chruściński对所有n×n复矩阵X和Y的广义Böttcher-Wenzel不等式提出了一些猜想：‖XY−YX‖1≤C‖X‖2‖Y‖3，其中‖⋅‖i （i=1,2,3）是Frobenius范数或ω-加权Frobenius范数。本文证明了X和Y是秩一矩阵时的猜想。

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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On some conjectures concerning the Böttcher-Wenzel inequality for weighted Frobenius norms

Let ω be a positive definite matrix. The ω-weighted Frobenius norm

{‖ \cdot ‖}_{ω}

is defined by

{‖ X ‖}_{ω} = \sqrt{tr ω X^{⁎} X}

. Recently, A. Mayumi, G. Kimura, H. Ohno, and D. Chruściński raised some conjectures concerning the generalized Böttcher-Wenzel inequality:

{‖ X Y - Y X ‖}_{1} \leq C {‖ X ‖}_{2} {‖ Y ‖}_{3} for all n \times n complex matrices X and Y,

where

{‖ \cdot ‖}_{i}

(

i = 1, 2, 3

) is the Frobenius norm or ω-weighted Frobenius norm. In this paper, the conjectures are proved when X and Y are rank one matrices.

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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.