Singular values and eigenvalues of matrices in \(\mathfrak{so}_{n}(\mathbb C)\) and \(\mathfrak{sp}_{n}(\mathbb C)\) (Q376640)

A theorem of Weyl and Horn states that, if \(A\) is a matrix from \({\mathbb C}_{n\times n}\) the set of all \(n\times n\) complex matrices with singular values \(s_1\geq s_2\geq \dots \geq s_n\) and eigenvalues \(\lambda_1, \lambda_2, \dots, \lambda_n\) ordered as \(|\lambda_1|\geq |\lambda_2| \geq \dots \geq |\lambda_n|\), then \(\prod _{j=1}^k|{\lambda _j}|\leq \prod _{j=1}^k{s_j}\), for \(k=1, \dots, n-1\) and \(\prod _{j=1}^n|{\lambda _j}|=\prod _{j=1}^n{s_j}\). Conversely, if \(s_1\geq s_2\geq \dots \geq s_n\) and \(|\lambda_1|\geq |\lambda_2| \geq \dots \geq |\lambda_n|\) satisfy \(\prod _{j=1}^k|{\lambda _j}|\leq \prod _{j=1}^k{s_j}\), for \(k=1, \dots, n-1\) and \(\prod _{j=1}^n|{\lambda _j}|=\prod _{j=1}^n{s_j}\), then there exists \(A\in {\mathbb C}_{n\times n}\) such that \(s_1, s_2, \dots, s_n\) and \(\lambda_1, \lambda_2, \dots, \lambda_n\) are respectively the singular values and eigenvalues of \(A\). The authors obtain analogues of this theorem for \(A\) taken from \(\mathfrak{so}_m({\mathbb C})\) the set of all \(m\times m\) complex skew-symmetric matrices and then from the symplectic algebras \(\mathfrak{sp}_n({\mathbb C})\) and \(\mathfrak{sp}_n({\mathbb R})\).