For every $2n\times 2n$ real positive definite matrix A, there exists a real symplectic matrix M such that $M^{T}AM=\mathrm{\text{diag}}(D,D)$, where D is the $n\times n$ positive diagonal matrix with diagonal entries $d_1(A)\le \cdots \le d_n(A)$. The numbers $d_1(A),\dots ,d_n(A)$ are called the symplectic eigenvalues of A. We derive analogues of Wielandt's extremal principle and multiplicative Lidskii's inequalities for symplectic eigenvalues.

中文翻译：

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辛特征值的和和乘积

对于每 $2n\times 2n$实正定矩阵A，存在实辛矩阵M使得${米}^{吨}\mathrm{一种}米=\mathrm{\text{诊断}}(D,D)$，其中D是$n\times n$ 具有对角元素的正对角矩阵 $d_1(\mathrm{一种})\le \cdots \le d_n(\mathrm{一种})$. 数字$d_1(\mathrm{一种}),\dots ,d_n(\mathrm{一种})$称为A的辛特征值。我们推导出 Wielandt 极值原理和辛特征值乘法 Lidskii 不等式的类似物。