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2009.77: Fiedler Companion Linearizations and the Recovery of Minimal Indices

2009.77: Fernando De Teran, Froilan M. Dopico and D. Steven Mackey (2009) Fiedler Companion Linearizations and the Recovery of Minimal Indices.

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Abstract

A standard way of dealing with a matrix polynomial $P(\lambda)$ is to convert it into an equivalent matrix pencil -- a process known as linearization. For any regular matrix polynomial, a new family of linearizations generalizing the classical first and second Frobenius companion forms has recently been introduced by Antoniou and Vologiannidis, extending some linearizations previously defined by Fiedler for scalar polynomials. We prove that these pencils are linearizations even when $P(\lambda)$ is a singular square matrix polynomial, and show explicitly how to recover the left and right minimal indices and minimal bases of the polynomial $P(\lambda)$ from the minimal indices and bases of these linearizations. In addition, we provide a simple way to recover the eigenvectors of a regular polynomial from those of any of these linearizations, without any computational cost. The existence of an eigenvector recovery procedure is essential for a linearization to be relevant for applications.

Item Type: MIMS Preprint singular matrix polynomials, matrix pencils, minimal indices, minimal bases, linearization, recovery of eigenvectors, Fiedler pencils, companion forms MSC 2000 > 15 Linear and multilinear algebra; matrix theoryMSC 2000 > 65 Numerical analysis 2009.77 Dr. D. Steven Mackey 31 March 2010