Some Convexity Features Associated with Unitary Orbits

[摘要] Let $mathcal{H}_n$ be the real linear space of $nimes n$ complexHermitian matrices. The unitary (similarity) orbit $mathcal{U} (C)$ of $C in mathcal{H}_n$ is the collection of all matricesunitarily similar to $C$. We characterize those $C in mathcal{H}_n$ such that every matrix in the convex hull of $mathcal{U}(C)$ canbe written as the average of two matrices in $mathcal{U}(C)$. Theresult is used to study spectral properties of submatrices ofmatrices in $mathcal{U}(C)$, the convexity of images of $mathcal{U}(C)$ under linear transformations, and some related questionsconcerning the joint $C$-numerical range of Hermitian matrices.Analogous results on real symmetric matrices are also discussed.

[学科分类] 数学（综合）