Zhang, D., Hou, L., Ma, L. (2017). Properties of matrices with numerical ranges in a sector. Bulletin of the Iranian Mathematical Society, 43(6), 1699-1707.

D. Zhang; L. Hou; L. Ma. "Properties of matrices with numerical ranges in a sector". Bulletin of the Iranian Mathematical Society, 43, 6, 2017, 1699-1707.

Zhang, D., Hou, L., Ma, L. (2017). 'Properties of matrices with numerical ranges in a sector', Bulletin of the Iranian Mathematical Society, 43(6), pp. 1699-1707.

Zhang, D., Hou, L., Ma, L. Properties of matrices with numerical ranges in a sector. Bulletin of the Iranian Mathematical Society, 2017; 43(6): 1699-1707.

Properties of matrices with numerical ranges in a sector

^{}Department of Mathematics, Shanghai University, Shanghai 200444, China.

Receive Date: 13 May 2016,
Revise Date: 21 September 2016,
Accept Date: 21 September 2016

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.

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