LINEARIZATIONS OF SINGULAR MATRIX POLYNOMIALS AND THE RECOVERY OF MINIMAL INDICES

By Fernando De Terán, Frolian Dopico, D. Steven Mackey, Mims EPrint. 2009

80 citations recorded by the source corpus.

A standard way of dealing with a regular matrix polynomial P (λ) 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 first and second companion forms have recently been introduced by Mackey, Mackey, Mehl and Mehrmann. Almost all of these pencils are linearizations for P (λ )w hen P is regular. The goal of this work is to show that most of the pencils in L1(P )a ndL2(P )a re stil l linearizations when P (λ) is a singular square matrix polynomial, and that these linearizations can be used to obtain the complete eigenstructure of P (λ), comprised not only of the finite and infinite 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 (λ )f rom the minimalindices and bases of l in L1(P )a ndL2(P ). As a consequence of the recovery formulae for minimal indices, we prove that the vector space DL(P )= L1(P ) ∩ L2(P )w il l never contain any linearization for a square singular polynomial P (λ). Finally, the results are extended to other linearizations of singular polynomials defined in terms of more general polynomial bases.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/bc98792d575f5ba0385e5dc9a6a862673d959a7c

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Citations in the corpus

Möbius transformations of matrix polynomials (2015, background): “However, unimodular equivalence can change any (even all) of the minimal indices, as illustrated in [14, 20, 21].”
Spectral equivalence of matrix polynomials and the index sum theorem (2014, background): “This in turn highlights the importance of results like those in [8, 9, 11], where minimal index recovery has been shown to be extremely simple for important classes of strong linearizations, and like those for the rectangular Frobenius-like companion forms of arbitrary grade introduced in Section 5 ”
Fiedler companion linearizations for rectangular matrix polynomials (2012, background): “At first, only linearizations for regular matrix polynomials were considered [1, 2, 23, 27, 28], while more recently square singular polynomials have also received attention [10, 11, 12, 34].”
A framework for analyzing nonlinear eigenproblems and parametrized linear systems (2011, background): “These two companion forms are always linearizations of P, even when P is not regular [9].”
Jordan structures of alternating matrix polynomials (2010, background): “This topic is currently under investigation in [5, 6], and will not be addressed in this paper.”
Smith Forms of Palindromic Matrix Polynomials (2011, unknown): “3 of [3], a regular pencil L(λ) is a strong linearization of a regular P (λ) if and only if L(λ) and P (λ) have the same finite and infinite elementary divisors.”