A parametrization of structure-preserving transformations for matrix polynomials

Abstract

Given a matrix polynomial A(λ) of degree d and the associated vector space of pencils DL(A) described in Mackey et al. [12], we construct a parametrization for the set of left and right transformations that preserve the block structure of such pencils. They form a special class of structure-preserving transformations (SPTs). An SPT in that class maps DL(A) to DL(A˜), where A˜(λ) is a new matrix polynomial that is still of degree d and whose finite and infinite eigenvalues and their partial multiplicities are the same as those of A(λ). Unlike previous work on SPTs, we do not require the leading matrix coefficient of A(λ) to be nonsingular. We show that additional constraints on the parametrization lead to SPTs that also preserve extra structures in A(λ) 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(λ) into a matrix polynomial A˜(λ) whose jth matrix coefficient A˜j is a low-rank modification of Aj. 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.