Matrix Polynomials with Completely Prescribed Eigenstructure

Abstract

Abstract. We present necessary and sufficient conditions for the existence of a matrix polynomial when its degree, its finite and infinite elementary divisors, and its left and right minimal indices are prescribed. These conditions hold for arbitrary infinite fields and are determined mainly by the "index sum theorem," which is a fundamental relationship between the rank, the degree, the sum of all partial multiplicities, and the sum of all minimal indices of any matrix polynomial. The proof developed for the existence of such polynomial is constructive and, therefore, solves a very general inverse problem for matrix polynomials with prescribed complete eigenstructure. This result allows us to fix the problem of the existence of -ifications of a given matrix polynomial, as well as to determine all their possible sizes and eigenstructures.

Polynomial zigzag matrices, dual minimal bases, and the realization of completely singular polynomials

“…Finally, in Section 7, we show how zigzag matrices can be used to provide simple, explicit constructions of polynomial matrices with any prescribed degree d, any prescribed lists of left and right minimal indices, and no elementary divisors at all (neither finite nor infinite), subject to the single necessary and sufficient condition that d divides the sum of all the prescribed minimal indices. The results in Section 7 complement results recently presented in [5], where a much more general inverse problem for matrix polynomials has been solved, but via a rather complicated construction which does not explicitly display the realized complete eigenstructure. We begin with a preliminary Section 2, where we remind the reader of a number of basic results that are needed throughout this work.…”

“…The Index Sum Theorem is an important result presented first for real polynomials in [16], and extended to polynomials over any field in [3]. Recently, it has been shown in [5,Remark 3.2] that the Index Sum Theorem is an easy corollary of a more general result valid for arbitrary rational matrices proved in [18,Theorem 3], much earlier than in the previous references. Theorem 2.13.…”

“…It was proved in [7, p. 503] by using minors of matrices. A different proof can be found in [5,Lemma 3.6], and a new proof is outlined in Remark 2.14.…”

“…The recent paper [5] has established very simple necessary and sufficient conditions for the existence of a polynomial matrix when its degree, finite and infinite elementary divisors, and left and right minimal indices are prescribed. The proof of these necessary and sufficient conditions (see Theorem 3.3 or the equivalent formulation Theorem 3.12 in [5]) combines the Index Sum Theorem (stated in Theorem 2.13) with the construction of a polynomial matrix P (λ) which realizes the prescribed structure and degree.…”

“…A proof of this result can be found in [7,Section 6]; see also [5,Lemma 3.6] for another proof based on techniques developed in [11,Chapter 6]. In this paper we study the associated inverse problem: given two lists of nonnegative integers (η 1 , η 2 , .…”