Publication: Polynomial zigzag matrices, dual minimal bases, and the realization of completely singular polynomials
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2016-01
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Elsevier
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Abstract
Minimal bases of rational vector spaces are a well-known and important tool in systems theory. If
minimal bases for two subspaces of rational n-space are displayed as the rows of polynomial matrices
Z1(λ)_(k x n) and Z2(λ)_(m x n), respectively, then z_1and z_2are said to be dual minimal bases if the subspaces have complementary dimension, i.e., k+m=n, and Z_1 (λ) Z_2^T (λ)=0. In other words, each z_j (λ) provides a minimal basis for the nullspace of the other. It has long been known that for any dual minimal bases
z_1 (λ) and z_2 (λ), the row degree sums of Z1 and Z2 are the same. In this paper we show that this is the
only constraint on the row degrees, thus characterizing the possible row degrees of dual minimal bases.
The proof is constructive, making extensive use of a new class of sparse, structured polynomial matrices
that we have baptized zigzag matrices. Another application of these polynomial zigzag matrices is the
constructive solution of the following inverse problem for minimal indices { given a list of left and right
minimal indices and a desired degree d, does there exist a completely singular matrix polynomial (i.e., a
matrix polynomial with no elementary divisors whatsoever) of degree d having exactly the prescribed
minimal indices? We show that such a matrix polynomial exists if and only if d divides the sum of the
minimal indices. The constructed realization is simple, and explicitly displays the desired minimal indices
in a fashion analogous to the classical Kronecker canonical form of singular pencils.
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Zigzag matrices, singular matrix polynomials, minimal indices, dual minimal bases, inverse problem.
Bibliographic citation
Linear Algebra and its Applications, 488 (2016) 1-January, pp. 460-504.