On the positive semidefinteness of a class of Hermitian Cauchy-like matrices

Abstract

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.