Symmetric Linearizations for Matrix Polynomials

Abstract

Abstract.A standard way of treating the polynomial eigenvalue problem P (λ)x = 0 is to convert it into an equivalent matrix pencil-a process known as linearization. Two vector spaces of pencils L 1 (P ) and L 2 (P ), and their intersection DL(P ), have recently been defined and studied by Mackey, Mackey, Mehl, and Mehrmann. The aim of our work is to gain new insight into these spaces and the extent to which their constituent pencils inherit structure from P . For arbitrary polynomials we show that every pencil in DL(P ) is block symmetric and we obtain a convenient basis for DL(P ) built from block Hankel matrices. This basis is then exploited to prove that the first deg(P ) pencils in a sequence constructed by Lancaster in the 1960s generate DL(P ). When P is symmetric, we show that the symmetric pencils in L 1 (P ) comprise DL(P ), while for Hermitian P the Hermitian pencils in L 1 (P ) form a proper subset of DL(P ) that we explicitly characterize. Almost all pencils in each of these subsets are shown to be linearizations. In addition to obtaining new results, this work provides a self-contained treatment of some of the key properties of DL(P ) together with some new, more concise proofs.

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

“…Thus pencils in DL(P) always have block symmetric coefficients, even when there is no structure in the matrix coefficients of P. What happens when P is structured? As shown in [38], when P is symmetric, the collection of all symmetric pencils in L 1 (P) is exactly DL(P), while for Hermitian P the Hermitian pencils in L 1 (P) form a proper (but nontrivial) subspace H(P) ⊂ DL(P).…”

“…An unexpected property of DL(P) itself was proved in [38]. Consider the block transpose of a block matrix, defined as follows.…”

“…An isomorphism between DL(P) and F k now follows, which in turn induces a natural basis for DL(P). Described in [38], a pencil λ X i + Y i in this basis has special structure. Every X i and Y i is block diagonal, with the diagonal blocks being blockHankel.…”

Numerical methods for palindromic eigenvalue problems: Computing the anti‐triangular Schur form

“…It has been shown in [15] how to obtain structure-preserving T -palindromic linearizations for T -palindromic polynomials, see also [7,19] for related results on structured linearizations. Recently, a new method was introduced in [3] for solving quadratic Tpalindromic eigenvalue problems via a structure-preserving doubling algorithm.…”

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

“…When working with matrix polynomials, it is standard to use a strong linearization to convert the polynomial eigenvalue problem to a generalized eigenvalue problem [3,14,15]. A strong linearization, as opposed to weaker linearizations, preserves eigenstructure at infinity as well as the finite eigenstructure.…”