Some counterexamples related to sectorial matrices and matrix phases

Examples and Counterexamples

Dan Wang¹  Axel Ringh²  Xin Mao³  Li Qiu³ 

[1] Department of Electronic and Computer Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China;Corresponding author.;Department of Electronic and Computer Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China;

关键词: Sectorial matrices;    Phases;    Majorization;    Schur complement;   

DOI  :  

来源: DOAJ

【 摘 要 】

A sectorial matrix is an n×nmatrix whose numerical range is contained in an open half-plane, and such matrices have many nice properties. In particular, the subset of strictly accretive matrices is a convex cone in the space of n×nmatrices, and results related to positive definite matrices have recently been generalized to this cone. Moreover, sectorial matrices have recently been used to define phases of a matrix, and these phases can be used to angularly bound the eigenvalues by majorization-type inequalities similar to the ones for the singular values and the absolute value of the eigenvalues. Nevertheless, many traits that would be desirable are not true for sectorial matrices and matrix phases, and in this note we present a number of counterexamples for such traits. More precisely, the counterexamples are related to sectorial polar decompositions, majorization inequalities for phases of products, the spectral and numerical radius, and Schur complements.

【 授权许可】

Unknown