Structured strong linearizations of structured rational matrices

TL;DR

This paper introduces a family of structure-preserving linearizations for structured rational matrices, enabling efficient computation of eigenvalues, eigenvectors, and indices while maintaining the matrices' inherent symmetries.

Contribution

It develops Fiedler-like pencils as strong linearizations that preserve the structure of rational matrices, including symmetry and Hamiltonian properties, and provides methods for eigenvector and index recovery.

Findings

Constructed structure-preserving strong linearizations for various matrix structures.

Showed that eigenvectors and indices can be recovered operation-free from linearizations.

Proved that transfer functions preserve the Cauchy-Maslov index for real symmetric cases.

Abstract

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational matrices have been introduced recently for computing poles, eigenvalues, eigenvectors, minimal bases and minimal indices of rational matrices. For structured rational matrices, it is desirable to construct structure-preserving linearizations so as to preserve the symmetry in the eigenvalues and poles of the rational matrices. With a view to constructing structure-preserving linearizations of structured rational matrices, we propose a family of Fiedler-like pencils and show that the family of Fiedler-like pencils is a rich source of structure-preserving strong linearizations of structured rational matrices. We construct symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, skew-Hermitian, para-Hermitian and para-skew-Hermitian strong linearizations of a rational matrix $G(\lambda)$ when $G(\lambda)$ has the same structure. Further, when $G(\lambda)$ is real and symmetric, we show that the transfer functions of real symmetric linearizations of $G(\lambda)$ preserve the Cauchy-Maslov index of $G(\lambda).$ We describe the recovery of eigenvectors, minimal bases and minimal indices of $G(\lambda)$ from those of the linearizations of $G(\lambda)$ and show that the recovery is operation-free.