L. Lerer

Minimality and realization of discrete time-varying systems

The minimality and realization theory is developed for discrete time-varying finite dimensional linear systems with time-varying state spaces. The results appear as a natural generalization of the corresponding theory for the time-independent case. Special attention is paid to periodical systems. The case when the state space dimensions do not change in time is re-examined.

Factored forms for solutions of AX − XB = C and X − AXB = C in companion matrices

Abstract The main concern of this paper is linear matrix equations with block-companion matrix coefficients. It is shown that general matrix equations AX − XB = C and X − AXB = C can be transformed to equations whose coefficients are block companion matrices: C L X−XC M = diag [I 0…0] and X− C L XC M = diag [I 0…0] , respectively, where ĈL and CM stand for the first and second block-companion matrices of some monic r × r matrix polynomials L(λ) = λsI + Σs−1j=0λjLj and M(λ) = λtI + Σt7minus;1j=0λjMj. The solution of the equat with block companion coefficients is reduced to solving vector equations Sx = ϱ, where the matrix S is r2l × r2l[l = max(s, t)] and enjoys some symmetry properties.

On minimality in the partial realization problem

Abstract A compression algorithm is constructed which allows one to reduce an arbitrary realization of a finite sequence of matrices M 1 ,…, M r to a minimal partial realization of M 1 ,…, M r . Furthermore, a state space criterion of minimality of a partial realization and a formula for the minimal state space dimension are obtained.

Minimal Divisors of Rational Matrix Functions with Prescribed Zero and Pole Structure

Necessary and sufficient conditions are given in order that a rational matrix function is a minimal divisor of another one. These conditions are expressed in terms of zero and pole structure of the given functions. In connection with this a description is obtained of all rational matrix functions with prescribed zero and pole data.

The Bezoutian and the eigenvalue-separation problem for matrix polynomials

A generalized Bezout matrix for a pair of matrix polynomials is studied and, in particular, the structure of its kernel is described and the relations to the greatest common divisor of the given matrix polynomials are presented. The classical root-separation problems of Hermite, Routh-Hurwitz and Schur-Cohn are solved for matrix polynomials in terms of this Bezout matrix. The eigenvalue-separation results are also expressed in terms of Hankel matrices whose entries are Markov parameters of rational matrix function. Some applications of Jacobis method to these problems are pointed out.

Generalized Bezoutian and the inversion problem for block matrices, I. General scheme

The unified approach to the matrix inversion problem initiated in this work is based on the concept of the generalized Bezoutian for several matrix polynomials introduced earlier by the authors. The inverse X−1 of a given block matrix X is shown to generate a set of matrix polynomials satisfying certain conditions and such that X−1 coincides with the Bezoutian associated with that set. Thus the inversion of X is reduced to determining the underlying set of polynomials. This approach provides a fruitful tool for obtaining new results as well as an adequate interpretation of the known ones.

Common multiples and common divisors of matrix polynomials, II. Vandermonde and resultant matrices

In this paper explicit formulas are given for least common multiples and greatest common divisors of a finite number of matrix polynomials in terms of the coefficients of the given polynomials. An important role is played by block matrix generalizations of the classical Vandermonde and resultant matrices. Special attention is given to the evaluation of the degrees and other characteristics. Applications to matrix polynomial equations and factorization problems are made.

Generalized Bezoutian and matrix equations

Abstract A natural generalization of the classical Bezout matrix of two polynomials is introduced for a family of several matrix polynomials. The main aim of the paper is to show that this generalized Bezoutian serves as an adequate connecting link between the class of equations in matrix polynomials M ( λ ) Y ( λ ) + Z ( λ ) L ( λ ) = R ( λ ) and the class of linear matrix equations AX − XB = C . Each equation in one of these classes is coupled with a certain equation in the other class so that for each couple the generalized Bezoutian corresponding to a solution ( Y ( λ ), Z ( λ )) of the equation in matrix polynomials is a solution of the matrix equation, and conversely, any solution X of the matrix equation is a generalized Bezoutian corresponding to a certain solution of the equation in matrix polynomials. In particular, either both equations are solvable or both have no solutions. Explicit formulas connecting the solutions of the two equations are given. Also, various representation formulas for the generalized Bezoutian are derived, and its relation to the resultant matrix and the greatest common divisor of several matrix polynomials is discussed.

Bezoutian and Schur-Cohn problem for operator polynomials

Abstract A generalized Bezout operator (Bezoutian) for a pair of operator polynomials is introduced and its kernel is described in terms of common spectral data of the underlying polynomials. The location of the spectrum of an operator polynomial with compact spectrum with respect to the unit circle (infinite-dimensional version of the Schur-Cohn problem) is expressed via the inertia of a suitable Bezoutian. An application to the geometric dichotomy problem for difference equations with operator coefficients is given as well.

Minimality and irreducibility of time-invariant linear boundary-value systems

Abstract Minimality and irreducibility in the class of time-invariant linear systems with well-posed boundary conditions are introduced and characterized. Necessary and sufficient conditions for similarity of irreducible (or minimal) time-invariant systems with well-posed boundary conditions are given. The problem of stable minimality is analysed. Some of the results are specified for stationary and displacement systems.