# Linearizations of Singular Matrix Polynomials and the Recovery of Minimal Indices

De Teran, Fernando and Dopico, Frolian M. and Mackey, D. Steven (2009) Linearizations of Singular Matrix Polynomials and the Recovery of Minimal Indices. [MIMS Preprint] There is a more recent version of this item available. PDF lin-sing-poly-revised_mims_18-09-2009_23-06-07.pdf Download (289kB) PDF lin-sing-poly-revised_mims.pdf Download (289kB)

## Abstract

A standard way of dealing with a regular matrix polynomial P(¸) is to convert it into an equivalent matrix pencil { a process known as linearization. Two vector spaces of pencils L1(P) and L2(P) that generalize the ¯rst and second companion forms have recently been introduced by Mackey, Mackey, Mehl and Mehrmann. Almost all of these pencils are linearizations for P(¸) when P is regular. The goal of this work is to show that most of the pencils in L1(P) and L2(P) are still linearizations when P(¸) is a singular square matrix polynomial, and that these linearizations can be used to obtain the complete eigenstructure of P(¸), comprised not only of the ¯nite and in¯nite eigenvalues, but also for singular polynomials of the left and right minimal indices and minimal bases. We show explicitly how to recover the minimal indices and bases of the polynomial P(¸) from the minimal indices and bases of linearizations in L1(P) and L2(P). As a consequence of the recovery formulae for minimal indices, we prove that the vector space DL(P) = L1(P) \ L2(P) will never contain any linearization for a square singular polynomial P(¸). Finally, the results are extended to other linearizations of singular polynomials de¯ned in terms of more general polynomial bases

Item Type: MIMS Preprint singular matrix polynomials, matrix pencils, minimal indices, minimal bases, linearization MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Ms Lucy van Russelt 21 Oct 2009 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/1251

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