## Singular values and eigenvalues of matrices in $$\mathfrak{so}_{n}(\mathbb C)$$ and $$\mathfrak{sp}_{n}(\mathbb C)$$.(English)Zbl 1280.15012

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 $$\mathfrak{sp}_n({\mathbb R})$$.

### MSC:

 15A42 Inequalities involving eigenvalues and eigenvectors 15A18 Eigenvalues, singular values, and eigenvectors 15B57 Hermitian, skew-Hermitian, and related matrices
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