## 2009.25: Linearizations of Singular Matrix Polynomials and the Recovery of Minimal Indices

2009.25:
Fernando De Teran, Frolian M. Dopico and D. Steven Mackey
(2009)
*Linearizations of Singular Matrix Polynomials and the Recovery of Minimal Indices.*

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## 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 |
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Uncontrolled Keywords: | singular matrix polynomials, matrix pencils, minimal indices, minimal bases, linearization |

Subjects: | MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2009.25 |

Deposited By: | Ms Lucy van Russelt |

Deposited On: | 21 October 2009 |

### Available Versions of this Item

- Linearizations of Singular Matrix Polynomials and the Recovery of Minimal Indices (deposited 22 October 2009)
- Linearizations of Singular Matrix Polynomials and the Recovery of Minimal Indices (deposited 21 October 2009)
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- Linearizations of Singular Matrix Polynomials and the Recovery of Minimal Indices (deposited 21 October 2009)

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