A Duality Reformulation of the Companion-Matrix Lyapunov Problem
Abstract
We study the relation between positive semidefiniteness and entrywise nonnegativity for solutions of continuous-time Lyapunov equations. For a real Hurwitz matrix A, we show that the solution of AP+PA⊤=−Q is positive semidefinite for every symmetric entrywise nonnegative Q if and only if the solution of A⊤ X+XA=−R is entrywise nonnegative for every positive semidefinite R. This equivalence follows from the adjointness of the two solution operators and extends to all real unmixed matrices. We then prove that both properties hold for every real Hurwitz companion matrix, settling a conjecture previously established under the additional assumption of a real spectrum. The proof combines a controllability-Gramian normalization with a pairwise positivity theorem for the coefficients of the adjugate polynomial of a real accretive matrix. The latter is obtained from a coefficient-sign property of bivariate polynomials, an auxiliary determinant that does not vanish on the product of two open right half-planes, and a rank-one perturbation argument. Covariance and energy interpretations connect these results with dissipative realizations, damped second-order systems, and comparisons between input Gramians.
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BibTeX
@misc{ferrante2026duality,
title = {{A Duality Reformulation of the Companion-Matrix Lyapunov Problem}},
author = {Augusto Ferrante},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13522},
eprint = {2609.13522},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13522},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Duality Reformulation of the Companion-Matrix Lyapunov Problem
AU - Augusto Ferrante
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.13522
UR - https://arxiv.org/abs/2609.13522
AB - We study the relation between positive semidefiniteness and entrywise nonnegativity for solutions of continuous-time Lyapunov equations. For a real Hurwitz matrix A, we show that the solution of AP+PA^⊤=−Q is positive semidefinite for every symmetric entrywise nonnegative Q if and only if the solution of A^⊤ X+XA=−R is entrywise nonnegative for every positive semidefinite R. This equivalence follows from the adjointness of the two solution operators and extends to all real unmixed matrices. We then prove that both properties hold for every real Hurwitz companion matrix, settling a conjecture previously established under the additional assumption of a real spectrum. The proof combines a controllability-Gramian normalization with a pairwise positivity theorem for the coefficients of the adjugate polynomial of a real accretive matrix. The latter is obtained from a coefficient-sign property of bivariate polynomials, an auxiliary determinant that does not vanish on the product of two open right half-planes, and a rank-one perturbation argument. Covariance and energy interpretations connect these results with dissipative realizations, damped second-order systems, and comparisons between input Gramians.
ER -