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PreprintCC BY-NC-ND 4.0arXiv

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

  • Augusto Ferrante
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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 −λ1,…,−λn, the matrix entries are defined through ratios of elementary symmetric polynomials in the variables λi. 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.

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Cite

BibTeXRIS
BibTeX
@misc{ferrante2026positive,
  title = {{On the positive semidefinteness of a class of Hermitian Cauchy-like matrices}},
  author = {Augusto Ferrante},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22550},
  eprint = {2609.22550},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22550},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - On the positive semidefinteness of a class of Hermitian Cauchy-like matrices
AU  - Augusto Ferrante
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.22550
UR  - https://arxiv.org/abs/2609.22550
AB  - 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,…,−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.
ER  -