Kronecker Invariants Research Articles

The rank distance problem for pairs of matrices and a completion of quasi-regular matrix pencils

In this paper we give a complete description of the possible feedback invariants of a pair of matrices submitted to an additive perturbation of low rank. Also, we describe the possible Kronecker invariants of a quasi-regular matrix pencil with a prescribed quasi-regular subpencil. All the results are valid over arbitrary fields.

Computing the extensions of preinjective and preprojective Kronecker modules

Let I, I′ be preinjective Kronecker modules (i.e. all their indecomposable components are preinjective). We describe the modules M for which there exists an exact sequence 0→I′→M→I→0 by explicit, easy to check numerical conditions, resulting in an algorithm (linear in the number of indecomposable components) for the decision problem. We also propose a method to generate all extensions of I′ by I and we give a different proof for a theorem in [13] providing numerical criteria in terms of Kronecker invariants for the existence of a monomorphism f:I′→I. All these results apply dually to preprojective modules as well.

Completion up to a matrix pencil with column minimal indices as the only nontrivial Kronecker invariants

In this paper we give explicit, necessary and sufficient conditions for a completion of an arbitrary matrix pencil up to a pencil with a prescribed set of Kronecker invariants, in the case when the only nontrivial invariants of the resulting pencil are its column (row) minimal indices. The result is given over arbitrary fields.

On preprojective short exact sequences in the Kronecker case

Let P , P ′ be preprojective Kronecker modules (i.e., all their indecomposable components are preprojective). We give a numerical criterion in terms of the so-called Kronecker invariants for the existence of an epimorphism g : P → P ′ . As an application, we describe the possible middle terms in certain preprojective short exact sequences. We also prove that the possible middle terms in preprojective short exact sequences do not depend on the base field k . All the results above can be dualized for preinjective modules.

Equivalence conditions for behaviors and the Kronecker canonical form

In this paper we explore equivalence conditions and invariants for behaviors given in kernel representations. In case the kernel representation is given in terms of a linear matrix pencil, the invariants for strict equivalence are given by the Kronecker canonical form which, in turn, we interpret in geometric control terms. If the behavior is given in a kernel representation by a higher order rectangular polynomial matrix, the natural equivalence concept is behavior equivalence. These notions are closely related to the Morse group that incorporates state space similarity transformations, state feedback, and output injection. A simple canonical form for behavioral equivalence is given that clearly exhibits the reachable and autonomous parts of the behavior. Using polynomial models we also present a unified approach to pencil equivalence that elucidates the close connections between classification problems from linear algebra, geometric control theory, and behavior theory. We also indicate how to derive the invariants under behavior equivalence from the Kronecker invariants.

Combinatorics of Column Minimal Indices and Matrix Pencil Completion Problems

In this paper we describe the possible Kronecker invariants of a matrix pencil with a prescribed subpencil, when the prescribed subpencil has only column (row) minimal indices as Kronecker invariants. This is done by exploiting combinatorics of column minimal indices and by introducing some novel labels. The result is given over arbitrary fields.

Singular systems, state feedback problem

In this paper the problem of Kronecker invariants assignment by state feedback in singular linear systems is studied and resolved. This result presents a generalization of the previous results of state feedback action on singular systems.

Formulas for Kronecker invariants using a representation theoretical approach

Using a representation theoretical modular approach we present some explicit formulas for the Kronecker invariants of a matrix pencil and we also give a short proof for the Demmel–Edelman codimension formula of a pencil.

Matrix pencils completion problems II

In this paper, we give new, simple and explicit necessary and sufficient conditions for the existence of a matrix pencil with prescribed Kronecker invariants and a regular subpencil. Moreover, we explicitly solve the problem of describing the possible Kronecker invariants of a pencil with a prescribed subpencil in the case of the existence of an intermediate regular pencil. All the results are obtained over algebraically closed fields.

DIRECTED GRAPHS AND KRONECKER INVARIANTS OF PAIRS OF MATRICES

Call two pairs (M,N) and (M′,N′) of m × n matrices over a field K, simultaneously K-equivalent if there exist square invertible matrices S,T over K, with M′ = SMT and N′ = SNT. Kronecker [2] has given a complete set of invariants for simultaneous equivalence of pairs of matrices. Associate in the natural way to a finite directed graph Γ, with v vertices and e edges, an ordered pair (M,N) of e × v matrices of zeros and ones. It is natural to try to compute the Kronecker invariants of such a pair (M,N), particularly since they clearly furnish isomorphism-invariants of Γ. Let us call two graphs "linearly equivalent" when their two corresponding pairs are simultaneously equivalent. There have existed, since 1890, highly effective algorithms for computing the Kronecker invariants of pairs of matrices of the same size over a given field [1,2,5,6] and in particular for those arising in the manner just described from finite directed graphs. The purpose of the present paper, is to compute directly these Kronecker invariants of finite directed graphs, from elementary combinatorial properties of the graphs. A pleasant surprise is that these new invariants are purely rational — indeed, integral, in the sense that the computation needed to decide if two directed graphs are linearly equivalent only involves counting vertices in various finite graphs constructed from each of the given graphs — and does not involve finding the irreducible factorization of a polynomial over K (in apparent contrast both to the familiar invariant-computations of graphs furnished by the eigenvalues of the connection matrix, and to the isomorphism problem for general pairs of matrices).

Matrix pencils completion problems

In this paper we give a partial solution to the challenge problem posed by Loiseau et al. in [J. Loiseau, S. Mondié, I. Zaballa, P. Zagalak, Assigning the Kronecker invariants of a matrix pencil by row or column completion, Linear Algebra Appl. 278 (1998) 327–336], i.e. we assign the Kronecker invariants of a matrix pencil obtained by row or column completion. We have solved this problem over arbitrary fields.

Subrepresentations of Kronecker representations

Translated into the language of representations of quivers, a challenge in matrix pencil theory is to find sufficient and necessary conditions for a Kronecker representation to be a subfactor of another Kronecker representation in terms of their Kronecker invariants. The problem is reduced to a numerical criterion for a Kronecker representation to be a subrepresentation of another Kronecker representation in terms of their Kronecker invariants. The key to the problem is the calculation of ranks of matrices over polynomial rings. For this, a generalization and specialization approach is introduced. This approach is applied to provide a numerical criterion for a preprojective (resp. regular, preinjective) Kronecker representation to be a subrepresentation of another preprojective (resp. regular, preinjective) Kronecker representation in terms of their Kronecker invariants.

Assigning the Kronecker invariants of a matrix pencil by row or column completions

The challenge consists in describing the relationships between the Kronecker invariants of a matrix pencil and one of its subpencils. For a given subpencil, an algorithm for constructing a matrix pencil with prescribed Kronecker invariants should also be proposed.

Kronecker bases for linear matrix equations, with application to two-parameter eigenvalue problems

The general solutions of the homogeneous matrix equation AXC T − BXD T = 0 and the system of the matrix equations AX + BY = 0, XC T + YD T = 0 are described in terms of Kronecker canonical forms, i.e., in terms of Kronecker invariants and Kronecker bases, for pairs of matrices ( A, B) and ( C, D). A canonical form for a pair of commuting matrices ( E, F) such that E 2 = F 2 = EF = 0 is discussed. These results are applied to construct a canonical basis for the second root subspace of a two-parameter eigenvalue problem. The corresponding relations for canonical invariants are given.

Grassmann invariants, matrix pencils, and linear system properties

The paper deals with the establishment of relationships between two different types of invariants defined on matrix pencils and polynomial matrices and highlights their significance in the context of linear systems. For general singular matrix pencil sF - G the Grassmann invariants of the pencil are introduced as the column, row Grassmann representatives g c( F, G), g r( F, G) t , and their corresponding Plücker matrices, P c( F, G), P r( F, G). The properties of these new invariants of matrix pencils are established. These results provide alternative means for classifying the different families of matrix pencils, which are important for the characterization of properties in linear systems. Finally, the relationship between the Grassmann-Plücker invariants of a general rational transfer function matrix and the system matrix pencil of a minimal realization is derived. The latter results provide means for the computation of transfer function Plücker matrices from state space descriptions, as well as enable the linking of state space Kronecker invariants to the Plücker invariants of transfer functions.

Structure Assignment of Right Invertible Systems Through Nonregular Feedback

Necessary and sufficient conditions for the characterization of possible modifications of the structure of right invertible linear systems through nonregular static feedback are obtained. The conditions are given in term of the Kronecker invariants of the open loop and closed loop system matrices. The problem is shown to be equivalent to a polynomial matrix completion problem.

The effect of sampling period on the behaviour of systems incorporating state feedback

An exposition of the effects of sampling on the controllability characteristics and the Kronecker invariants of the plant and the input matrices of sampled-data systems is presented. In particular, the effect of sampling period on the feedback matrix for time-optimal control of sampled-data systems is investigated in detail, and it is shown that different classes of systems produce different response characteristics as the sampling period becomes large. Also, simple expressions for A(NT) in terms of A(T), and B(NT) as a function of A(T) and B(T), are derived. These expressions are computationally attractive and therefore facilitate the study of the effect of sampling period on system behaviour. By applying these expressions, the effects of sampling period on the feedback matrices for time-optimal control of three classes of systems whose open-loop plants are asymptotically stable, marginally stable, and unstable, are illustrated by numerical examples.

Decomposability and Quotient Subspaces for the Pencil $sL - M$

This paper introduces the notion of decomposability of the domain and the codomain relative to the generalized nonsquare matrix pencil $\Pi ( s ) = sL - M$. Its importance is justified rigorously and it is demonstrated that a special case of these new results is the familiar notion of decomposability for the pencil $sI - M$. This definition is motivated by the concepts of strict equivalence and quotient subspaces. By decomposing both the domain and the codomain of L and M the close relation between decomposability and the Kronecker invariants of the pencil $sL - M$ is shown. Finally, an application of the notion of decomposabilty in controls design is presented.

Parametric time-optimal control of linear discrete-time systems by state feedback Part 2. Irregular Kronecker invariants

In an earlier paper (Part 1), a parametric state feedback time-optimal control law was devised for the regular Kronecker invariants on the basis of parametrization of the vector companion form. The present paper describes a straight-forward technique for obtaining the parametric feedback matrix by exploiting the parametric vector companion forms in the case of irregular Kronecker invariants. Hidden properties of vector companion forms are also given which result in entirely different and interesting techniques for parametrization of the control law. An illustrative example is presented.

Parametric time-optimal control of linear discrete-time systems by state feedback Part 1. Regular Kronecker invariants

Parametric time-optimal control of linear discrete-time systems by state feedback Part 1. Regular Kronecker invariants