Properties of matrices with numerical ranges in a sector

Document Type: Research Paper

Authors

Department of Mathematics‎, ‎Shanghai University‎, ‎Shanghai 200444‎, ‎China.

Abstract

Let $(A)$ be a complex $(n\times n)$ matrix and assume that the numerical range of $(A)$ lies in the set of a sector of half angle $(\alpha)$ denoted by $(S_{\alpha})$. We prove the numerical ranges of the conjugate, inverse and Schur complement of any order of $(A)$ are in the same $(S_{\alpha})$.
The eigenvalues of some kinds of matrix product and numerical ranges of hadmard product, star-congruent matrix and unitary matrix of polar decompostion are also included in the same sector. Furthermore, we extend some inequalities about eigenvalues and singular values and the linear fractional maps to this class of matrices.

Keywords

Main Subjects


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Volume 43, Issue 6
November and December 2017
Pages 1699-1707
  • Receive Date: 13 May 2016
  • Revise Date: 21 September 2016
  • Accept Date: 21 September 2016