一类Hermitian Cauchy-like矩阵的正半定性

中文摘要 AI 辅助

我们建立了一类与具有不同零点的实Hurwitz多项式相关的Hermitian Cauchy-like矩阵的正半定性。将零点记为$-\lambda_1,\ldots,-\lambda_n$，矩阵元素通过变量$\lambda_i$的初等对称多项式的比值定义。该结果扩展了先前在假设所有零点均为实数的情况下获得的正定性定理。我们首先证明分母中出现的系数非零，从而确保矩阵定义良好。然后，我们构造每个矩阵与一个状态矩阵为友矩阵形式的Lyapunov方程的解之间的显式合同变换。正半定性由关于此类具有逐项非负右端数据的方程的一个近期结果得出。该方法通过结构化矩阵与Lyapunov方程之间的联系，将结论扩展到复零点情形。

英文摘要

We establish the positive semidefiniteness of a class of Hermitian Cauchy-like matrices associated with real Hurwitz polynomials having distinct zeros. Writing the zeros as $-λ_1,\ldots,-λ_n$, the matrix entries are defined through ratios of elementary symmetric polynomials in the variables $λ_i$. The result extends an earlier positivity theorem obtained under the assumption that all zeros are real. We first show that the coefficients appearing in the denominators are nonzero, so that the matrices are well defined. We then construct an explicit congruence between each matrix and the solution of a Lyapunov equation whose state matrix is in companion form. Positive semidefiniteness follows from a recent result on such equations with entrywise nonnegative right-hand-side data. This approach establishes the extension to complex zeros through the connection between structured matrices and Lyapunov equations.

发表机构

• Università di Padova（帕多瓦大学）

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