矩阵乘积与数值域性质的推广

Generalization a Property of the Numerical Range and Product of Matrices

设A为一个n×n矩阵,σ(A),W(A)分别表示A的谱集和数值域.若对任意n×n酉矩阵U有σ(AU)■W(A)W(U),那么A为数乘半正定矩阵,从而改进和推广了一些相关的结果.

Let A be an n ×n matrix, and let a（A）,W（A） denote respectively the spectrum and the numerical range of A. It is shown that A is a scalar multiple of a positive semidefinite matrix if the inclusion a （AU） W （A）W（U） holds for any n × n unitary matrix U. Therefore, some classic results are generalized and improved.