A Parametrization of Structure-Preserving Transformations for Matrix Polynomials

Given a matrix polynomial $A(\lambda)$ of degree $d$ and the associated vector space of pencils $\DLP(A)$ described in Mackey, Mackey, Mehl, and Mehrmann [SIAM J. Matrix Anal. Appl., 28 (2006), pp. 971-1004], we construct a parametrization for the set of left and right transformations that preserve the block structure of such pencils, and hence produce a new matrix polynomial $\At(\lambda)$ that is still of degree $d$ and is unimodularly equivalent to $A(\lambda)$. We refer to such left and right transformations as structure-preserving transformations (SPTs). Unlike previous work on SPTs, we do not require the leading matrix coefficient of $A(\lambda)$ to be nonsingular. We show that additional constraints on the parametrization lead to SPTs that also preserve extra structures in $A(\lambda)$ such as symmetric, alternating, and $T$-palindromic structures. Our parametrization allows easy construction of SPTs that are low-rank modifications of the identity matrix. The latter transform $A(\lambda)$ into an equivalent matrix polynomial $\At(\lambda)$ whose $j$th matrix coefficient $\At_j$ is a low-rank modification of $A_j$. We expect such SPTs to be one of the key tools for developing algorithms that reduce a matrix polynomial to Hessenberg form or tridiagonal form in a finite number of steps and without the use of a linearization