Some Convexity Features Associated with Unitary Orbits

Choi, Man-Duen ; Li, Chi-Kwong ; Poon, Yiu-Tung

Canadian journal of mathematics, Tome 55 (2003) no. 1, pp. 91-111

Résumé

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.

Détail

DOI : 10.4153/CJM-2003-004-x

Mots-clés : 15A60, 15A42, Hermitian matrix, unitary orbit, eigenvalue, joint numerical range

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