Geometry of Matrix Polynomial Spaces

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-019-09423-1 Geometry of Matrix Polynomial Spaces Andrii Dmytryshyn1,2 · Stefan Johansson2 · Bo Kågström2 · Paul Van Dooren3 Received: 27 March 2018 / Revised: 5 February 2019 / Accepted: 29 March 2019 © The Author(s) 2019 Abstract We study how small perturbations of general matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy (stratification) graphs of matrix polynomials’ orbits and bundles. To solve this problem, we construct the stratification graphs for the first companion Fiedler linearization of matrix polynomials. Recall that the first companion Fiedler linearization as well as all the Fiedler linearizations is matrix pencils with particular block structures. Moreover, we show that the stratification graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler linearizations have the same geometry (topology). This geometry coincides with the geometry of the space of matrix polynomials. The novel results are illustrated by examples using the software tool StratiGraph extended with associated new functionality. Keywords Matrix polynomials · Stratifications · Matrix pencils · Fiedler linearization · Canonical structure information · Orbit · Bundle Mathematics Subject Classification 15A21 · 15A22 · 65F15 · 47A07 Communicated by Alan Edelman. Preprint Report UMINF 15.17 (revised), Department of Computing Science, Umeå University. B Andrii Dmytryshyn ; Stefan Johansson Bo Kågström Paul Van Dooren 1 School of Science and Technology, Örebro University, 701 82 Örebro, Sweden 2 Department of Computing Science, Umeå University, 901 87 Umeå, Sweden 3 Department of Mathematical Engineering, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium 123 Foundations of Computational Mathematics 1 Introduction For a long-time matrix polynomials P(λ) = λd Ad + · · · + λA1 + A0 , Ai ∈ Cm×n , i = 0, . . . , d, and Ad = 0, (1) have been important objects to investigate. Due to challenging applications [27,28,37, 41,42], matrix polynomials have received much attention in the last decade, resulting in rapid developments of corresponding theories [5–7,19,32,37] and computational techniques [3,27,34,36,39] (see also the recent survey [38]). In a number of cases, the canonical structure information, i.e. elementary divisors and minimal indices of the matrix polynomials, are the actual objects of interest. This information is usually computed via linearizations [3], in particular, Fiedler linearizations [1], i.e. matrix polynomials of degree d = 1 which are matrix pencils with a particular block structure. However, the canonical structure information is sensitive to perturbations in the coefficient matrices of the polynomial. How small perturbations may change the canonical structure information can be studied through constructing the orbit and bundle closure hierarchy (or stratification) graphs. Each node of such a graph represents a set of matrix polynomials with a certain canonical structure information, and there is an edge from one node to another if we can perturb any matrix polynomial associated with the first node such that its canonical structure information becomes equal to one of the matrix polynomials associated with the second node. The theory to compute and construct the stratification graphs is already known for several matrix problems: matrices under similarity (i.e. Jordan canonical form) [4,21,35,40], matrix pencils (i.e. Kronecker canonical form) [21], skew-symmetric matrix pencils [16], controllability and observability pairs [22], state-space system pencils [15], as well as full (normal)-rank matrix polynomials [32]. Many of these results are already implemented in StratiGraph [29,31,33], which is a java-based tool developed to construct and visualize such closure hierarchy graphs. The Matrix Canonical Structure (MCS) Toolbox for MATLAB [14,29,31] was also developed for simplifying the work with the matrices in canonical forms and connecting MATLAB with StratiGraph. For more details on each of these cases, we recommend to check the corresponding papers and their references; some control applications are discussed in [33]. In this paper, we study how small perturbations of general matrix polynomials, with rectangular matrix coefficients, may change their elementary divisors and minimal indices by constructing the closure hierarchy graphs of the orbits and bundles of matrix polynomial and their Fiedler linearizations. Our new results generalize and extend results from [32], where the study concerned full-rank matrix polynomials. Other recent results that are crucial for this study include necessary and sufficient conditions for a matrix polynomial with certain degree and canonical structure information to exist [7]; the strong linearization templates and how the minimal indices of such linearizations are related to the minimal indices of the polynomials [6]; the correspondence between perturbations of the linearizations and perturbations of matrix polynomials [32]; as well as the algorithm for the stratification of general matrix pencils [21]. In particular, the results in [6] and [7] allow us to consider polynomials with both left and right minimal indices, in contrast to [32] (recall that full-rank matrix 123 Foundations of Computational Mathematics polynomials may have either left or right minimal indices, not both types); as well as to use any Fiedler linearization in contrast to the fixed choice of either the first or second companion forms (depending on which type of the minimal indices is present). The rest of the paper is organized as follows. Sections 2–5 present necessary background to matrix polynomials, their linearizations and perturbations, and to matrix pencils. Codimension computation is presented in Sect. 6. Section 7 is devoted to stratifications of Fiedler linearizations of matrix polynomials. Section 7.1 recalls cover relations for complete eigenstructures, a concept frequently used in the results that follow on neighbours in the stratifications. Sections 7.2 and 7.3 provide the results for neighbouring orbits and bundles, respectively. All results are illustrated with examples. Finally, in Sect. 8 stratification results from Sect. 7 are expressed in terms of matrix polynomial invariants. Altogether, we complete the stratification theory for general matrix polynomials and the associated Fiedler linearizations. All matrices that we consider have complex entries. 2 Matrix Polynomials with Prescribed Invariants In this section, we consider matrix polynomials (1) and recall the definitions of the canonical structure information for matrix polynomials, i.e. the elementary divisors and minimal indices, and state Theorem 2 (proven in [7]) that explains which canonical structure information a matrix polynomial may have. Definition (...truncated)