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Some Convexity Features Associated with Unitary Orbits

Published online by Cambridge University Press:  20 November 2018

Man-Duen Choi ,
Chi-Kwong Li  and
Yiu-Tung Poon
Man-Duen Choi
Affiliation:
Department of Mathematics, University of Toronto Toronto, Ontario, M5S 3G3, e-mail: [email protected]
Chi-Kwong Li
Affiliation:
Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187, U.S.A., e-mail: [email protected]
Yiu-Tung Poon
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011, U.S.A., e-mail: [email protected]
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Abstract

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Let ${{\mathcal{H}}_{n}}$ be the real linear space of $n\,\times \,n$ complex Hermitian matrices. The unitary (similarity) orbit $\mathcal{U}\left( C \right)$ of $C\,\in \,{{\mathcal{H}}_{n}}$ is the collection of all matrices unitarily similar to $C$. We characterize those $C\,\in \,{{\mathcal{H}}_{n}}$ such that every matrix in the convex hull of $\mathcal{U}\left( C \right)$ can be written as the average of two matrices in $\mathcal{U}\left( C \right)$. The result is used to study spectral properties of submatrices of matrices in $\mathcal{U}\left( C \right)$, the convexity of images of $\mathcal{U}\left( C \right)$ under linear transformations, and some related questions concerning the joint $C$-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.

Keywords

15A6015A42Hermitian matrixunitary orbiteigenvaluejoint numerical range
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 1 , 01 February 2003 , pp. 91 - 111
Copyright
Copyright © Canadian Mathematical Society 2003

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