Invariant subspaces of $T$-palindromic pencils and algebraic $T$-Riccati equations

TL;DR

This paper explores the connection between algebraic $ op$-Riccati equations and deflating subspaces of $ op$-palindromic matrix pencils, providing new theoretical conditions and improved computational algorithms.

Contribution

It introduces conditions to avoid eigenvalues of modulus one and new algorithms for computing deflating subspaces, enhancing existing methods.

Findings

Conditions to prevent modulus-one eigenvalues in $ op$-palindromic pencils

New algorithms for deflating subspace computation

Improved palindromic QZ algorithm with a novel ordering procedure

Abstract

By exploiting the connection between solving algebraic $\top$-Riccati equations and computing certain deflating subspaces of $\top$-palindromic matrix pencils, we obtain theoretical and computational results on both problems. Theoretically, we introduce conditions to avoid the presence of modulus-one eigenvalues in a $\top$-palindromic matrix pencil and conditions for the existence of solutions of a $\top$-Riccati equation. Computationally, we improve the palindromic QZ algorithm with a new ordering procedure and introduce new algorithms for computing a deflating subspace of the $\top$-palindromic pencil, based on quadraticizations of the pencil or on an integral representation of the orthogonal projector on the sought deflating subspace.