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# Some Convexity Features Associated with Unitary Orbits

Published:2003-02-01
Printed: Feb 2003
• Man-Duen Choi
• Chi-Kwong Li
• Yiu-Tung Poon
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## Abstract

Let $\mathcal{H}_n$ be the real linear space of $n\times n$ complex Hermitian matrices. The unitary (similarity) orbit $\mathcal{U} (C)$ 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 $\mathcal{U}(C)$ can be written as the average of two matrices in $\mathcal{U}(C)$. The result is used to study spectral properties of submatrices of matrices in $\mathcal{U}(C)$, the convexity of images of $\mathcal{U} (C)$ 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: Hermitian matrix, unitary orbit, eigenvalue, joint numerical range
 MSC Classifications: 15A60 - Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] 15A42 - Inequalities involving eigenvalues and eigenvectors

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