## 2010.33: Palindromic Companion Forms for Matrix Polynomials of Odd Degree

2010.33:
Fernando De Teran, Froilan M. Dopico and D. Steven Mackey
(2010)
*Palindromic Companion Forms for Matrix Polynomials
of Odd Degree.*

*This is the latest version of this eprint.*

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## Abstract

The standard way to solve polynomial eigenvalue problems $P(\la)x=0$ is to convert the matrix polynomial $P(\la)$ into a matrix pencil that preserves its spectral information -- a process known as linearization. When $P(\la)$ is palindromic, the eigenvalues, elementary divisors, and minimal indices of $P(\la)$ have certain symmetries that can be lost when using the classical first and second Frobenius companion linearizations for numerical computations, since these linearizations do not preserve the palindromic structure. Recently new families of pencils have been introduced with the goal of finding linearizations that retain whatever structure the original $P(\la)$ might possess, with particular attention to the preservation of palindromic structure. However, no general construction of palindromic linearizations valid for all palindromic polynomials has as yet been achieved. In this paper we present a family of linearizations for odd degree polynomials $P(\la)$ which are palindromic whenever $P(\la)$ is, and which are valid for all palindromic polynomials of odd degree. We illustrate our construction with several examples. In addition, we establish a simple way to recover the minimal indices of the polynomial from those of the linearizations in the new family.

Item Type: | MIMS Preprint |
---|---|

Uncontrolled Keywords: | matrix polynomial, matrix pencil, Fiedler pencils, palindromic, companion form, minimal indices, structured linearization |

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

MIMS number: | 2010.33 |

Deposited By: | Dr. D. Steven Mackey |

Deposited On: | 23 June 2011 |

### Available Versions of this Item

- Palindromic Companion Forms for Matrix Polynomials
of Odd Degree (deposited 23 June 2011)
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