Independence of eigenvalues and independence of singular values of submatrices (Q1185054)

Let \(H\) be an \(n\times n\) Hermitian matrix and let \(2\leq k\leq n\), \(1\leq r\leq n/2\) be fixed with \(kr\leq n\). The author shows that as \(U\) varies over all unitary matrices, the eigenvalues of certain principal submatrices of \(UHU^*\) independently assume all values permitted by certain interlacing inequalities if and only if each distinct eigenvalue of \(H\) has multiplicity at least \(rk\). This generalizes a result of \textit{R. C. Thompson} [ibid. 2, 355-374 (1969; Zbl 0224.15014)]. The author also gives some necessary conditions for the independence of singular values of certain principal submatrices of \(UAV\), where \(A\) is a complex matrix and \(U\) and \(V\) vary over all unitary matrices. In some cases these conditions are also proved to be sufficient.