On minor prime factorizations for multivariate polynomial matrices

Multidimensional Systems and Signal Processing - Multivariate polynomial matrix factorizations have been widely investigated during the past years due to the fundamental importance in the areas of...     

On factor left prime factorization problems for multivariate polynomial matrices

This paper is concerned with factor left prime factorization problems for multivariate polynomial matrices without full row rank. We propose a necessary and sufficient condition for the existence of factor left prime factorizations of a class of multivariate polynomial matrices, and then design an algorithm to compute all factor left prime factorizations if they exist. We implement the algorithm on the computer algebra system Maple, and two examples are given to illustrate the effectiveness of the algorithm. The results presented in this paper are also true for the existence of factor right prime factorizations of multivariate polynomial matrices without full column rank.

     

Factorizations for a class of multivariate polynomial matrices

Multidimensional Systems and Signal Processing - This paper investigates how to factorize a class of multivariate polynomial matrices. We prove that an $$l\times m$$ multivariate polynomial matrix...     

Direct synthesis of discrete-time feedback systems by equivalence transformations of polynomial matrices

It is shown that the synthesis of closed-loop linear multivariable discrete-time systems can be directly effected by performing equivalence transformations on appropriate polynomial matrices. These polynomial matrices are the Smith canonical forms of the closed-loop characteristic matrices of such systems subject to the constraints imposed by the fundamental theorem of linear state-variable feedback.     

Relatively prime polynomial matrices

Relatively prime polynomial matrices are defined, and some of their properties are deduced and applied. The problem of extending these properties is posed.     

Factorizations fornD polynomial matrices

In this paper, a constructive general matrix factorization scheme is developed for extracting a nontrivial factor from a givennD (n>2) polynomial matrix whose maximal order minors satisfy certain conditions. It is shown that three classes ofnD polynomial matrices admit this kind of general matrix factorization. It turns out that minor prime factorization and determinantal factorization are two interesting special cases of the proposal general factorization. As a consequence, the paper provides a partial solution to an open problem of minor prime factorization as well as to a recent conjecture on minor prime factorizability fornD polynomial matrices. Three illustrative examples are worked out in detail.     

Inversion of rational polynomial matrices

The letter is concerned with automatic computational procedures for the inversion of a matrix where every element is a rational function. Two possible methods are considered: one based on the Gaussian elimination algorithm and the other based on the Faddeev algorithm.     

Generation of polynomial system matrices

It is shown that certain polynomial system matrices giving rise to a given transfer-function matrix can all be generated by a defined transformation from a representation of least order.     

Detecting infinite zeros in polynomial matrices

A simple necessary and sufficient algebraic condition is given to detect zeros at infinity in a polynomial matrix.     

Spectral factorization of two-variable para-Hermitian polynomial matrices

This paper presents an algorithm for the so-called spectral factorization of two-variable para-Hermitian polynomial matrices which are nonnegative definite on thej axis, arising in the synthesis of two-dimensional (2-D)passive multiports, Wiener filtering of 2-D vector signals, and 2-D control systems design. First, this problem is considered in the scalar case, that is, the spectral factorization of polynomials is treated, where the decomposition of a two-variable nonnegative definite real polynomial in a sum of squares of polynomials in one of the two variables having rational coefficients in the other variable plays an important role (cf. Section 4). Second, by using these results, the matrix case can be accomplished, where in a first step the problem is reduced to the factorization of anunimodular para-Hermitian polynomial matrix which is nonnegative definite forp=j , and in a second step this simplified problem is solved by using so-called elementary row and column operations which are based on the Euclidian division algorithm. The matrices considered may be regular or singular and no restrictions are made concerning the coefficients of their polynomial entries; they may be either real or complex.     

Some properties of relatively prime polynomial matrices

Some necessary and sufficient conditions are given for the polynomial matrices sIn, ? A and B to be relatively prime.     

Generalized inverses of two-variable polynomial matrices and applications

The main contribution of this paper is to present (a) an algorithm for the computation of the generalized inverse of a not necessarily square two-variable polynomial matrix and (b) some applications of the proposed algorithm to the solution of Diophantine equations.This work is supported by the Greek General Secretariat of Industry, Research and Technology.     

Bounds for the degrees of the entries of left inverses of polynomial matrices

We provide explicit bounds for the degrees of the polynomials which appear as the entries of the left inverse of a polynomial matrixF. When such an inverse does not exist, bounds can be given for the entries of the (1)-inverse ofF.     

On the ranks of certain matrices

A few basic results regarding the ranks of certain types of matrices are presented. Also, some bounds on the ranks of the real and imaginary parts of a positive semi-definite Hermitian matrix are worked out. Applications of particular cases of these results in the areas like passive network synthesis, generation of two-variable reactance matrices, etc., are pointed out.     

On the computation of discrete Legendre polynomial coefficients

A new and fast method to find the discrete Legendre polynomial (DLP) coefficients is presented. The method is based on forming a simple matrix using addition only and then multiplying two elements of the matrix to compute the DLP coefficients.     

On the equivalence between SOVA and max-log-MAP decodings

It is shown that after a proper simple modification, the soft-output Viterbi algorithm (SOVA) proposed by Hagenauer and Hoeher (1989) becomes equivalent to the max-log-maximum a posteriori (MAP) decoding algorithm. Consequently, this modified SOVA allows to implement the max-log-MAP decoding algorithm by simply adjusting the conventional Viterbi algorithm. Hence, it provides an attractive solution to achieve low-complexity near-optimum soft-input soft-output decoding     

On the equivalence of set-theoretic and maxent MAP estimation

We establish an equivalence between two conceptually different methods of signal estimation under modeling uncertainty, viz., set-theoretic (ST) estimation and maximum entropy (maxent) MAP estimation. The first method assumes constraints on the signal to be estimated, and the second assumes constraints on a probability distribution for the signal. We provide broad conditions under which these two estimation paradigms produce the same signal estimate. We also show how the maxent formalism can be used to provide solutions to three important problems: how to select sizes of constraint sets in ST estimation (the analysis highlights the role of shrinkage); how to choose the values of parameters in regularized restoration when using multiple regularization functionals; how to trade off model complexity and goodness of fit in a model selection problem.     

On the Newton bivariate polynomial interpolation with applications

The main purpose of this work is to provide recursive algorithms for the computation of the Newton interpolation polynomial of a given two-variable function. The special case where the interpolation polynomial has known upper bounds on the degree of each indeterminate is studied and applied to the computation of the inverse of a two-variable polynomial matrix.     

On Schur stable multivariate polynomials

The class of stable multivariate polynomials, recently introduced by Kaczorek (1985), is the largest class of polynomials preserving the stability property under small coefficient variations. The principal goal of the contribution is to show that the class of Schur stable multivariate polynomials is the Moebius transformation of the latter one. This fundamental relation provides a vehicle to translate results known for one class to the other one.