Determinant of block-Toeplitz band matrices

AbstractSome expressions are given for the determinant of an mn×mn block-Toeplitz band matrix L=[Li−j], with bandwidth (p+q+1)n<mn, in terms of the n×n generating matrix polynomial L(λ)=Σp+qj=0λjLp−j, detL-q≠0. In the scalar case this yields formulas for the determinant expressed via the zeros of the generating (scalar) polynomial. The approach adopted in this work leans heavily on the recently developed spectral theory of matrix polynomials

Generalized Bezoutian and matrix equations

AbstractA 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

smt: a Matlab structured matrices toolbox

We introduce the smt toolbox for Matlab. It implements optimized storage and fast arithmetics for circulant and Toeplitz matrices, and is intended to be transparent to the user and easily extensible. It also provides a set of test matrices, computation of circulant preconditioners, and two fast algorithms for Toeplitz linear systems.Comment: 19 pages, 1 figure, 1 typo corrected in the abstrac