Interpolation algorithm for computing Drazin inverse of polynomial matrices

A study on new right/left inverses of nonsquare polynomial matricesThis paper presents several new results on the inversion of full normal rank nonsquare polynomial matrices. New analytical right/left inverses of polynomial matrices are introduced, including the so-called τ-inverses, σ-inverses and, in particular,S-inverses, the latter providing the most general tool for the design of various polynomial matrix inverses. The applicationoriented problem of selecting stable inverses is also solved. Applications in inverse-model control, in particular robust minimum variance control, are exploited, and possible applications in signal transmission/recovery in various types of MIMO channels are indicated.

This paper gives an algorithm for computing the inverses of polynomial matrices. The Gauss-Jordan inversion method, which is commonly used for the numerical inversion of constant matrices, can be applied to the inversion of polynomial matrices. It requires, however, operations with polynomials. Moreover, in the division of the operations, if the common factors in the divisor polynomials and the dividend polynomials are not eliminated, the resultant inverse contains polynomials of high degree in the numerators and the denominators. An algorithm given in this paper requires only operations with constant matrices. This algorithm gives an inverse in the minimal degree form if a polynomial matrix to be inverted is not a special form. It is shown that this is faster than existing ones. Also, it is an extension of the Souriau-Frame-Faddeev one. Some examples are illustrated to show the feasibility of the algorithm.

An algorithm for inverting rational matrices

The main purpose of this paper is to present a quicker and less memory-expensive algorithm for the generalized inversion of polynomial matrices than those presented earlier (Karampetakis, 1997a Computation of the generalized inverse of a polynomial matrix and applications. Linear Algebr. Appl.252, 35–60 and Karampetakis, 1997b Generalized inverses of two variable polynomial matrices and applications. Circuit Syst. & Signal Process.16, 439–453).

Inversion components of block Hankel-like matrices

On Bezoutians of nonsquare matrix polynomials and inversion of matrices with nonsquare blocks

Let P be an arbitrary field and let P(x) be a ring of polynomials over P. Denote by Pn and Pn (x) the rings of n xn matrices over P and P(x), respectively. Consider a nonsingular polynomial matrix A(x) e P~ (x) (detA(x) $ 0) and represent it as a matrix polynomial over the field P: is a unital matrix polynomial. In this paper, we establish conditions under which a unital divisor D (x) extracted from the matrix polynomial A (x) is determined uniquely by its characteristic polynomial detA (x) = d(x). The re- sult obtained enables one to indicate the class of matrix polynomials for which this unital divisor is unique for a given characteristic polynomial. We also show how this result can be used when solving matrix polynomial equations. The uniqueness of a tmital divisor for a matrix polynomial over an algebraically closed field with char- acteristic zero was studied in ( 1 ). It is known (2) that, for a matrix A (x) e Pn (x), there exist matrices U(x), V(x) ~ GL n (Pn (x)) such that U (x)A (x) V(x) = F A (X) = diag (al(x), a2(x) ..... an(x)), where aj(x) e P(x), j= 1, 2 ..... n, are unital polynomials and ai(x ) (ai+l(x), i = 1, 2 ..... n - 1. The matrix FA(X ) is called a canonical diagonal form of the matrix A(x). It is also known that if a matrix is nonsingular, then it can be reduced to the normal Hermite form by right elementary transformations, i.e., for the matrix A (x), there exists a matrix W(x) e GLn (P(x)) such that

The features of generating the n th sequence of refined Fibonacci polynomial matrices of the m th order, the elements of which are Fibonacci polynomials of no higher than (m+n–2)th number, are considered. The obtained Fibonacci matrices allow finding both their determinants and inverse matrices suitable for matrix encryption of block data. It was found that even over the last decade a significant number of publications have been published, each of which substantiates different approaches to generating sequences of Fibonacci polynomial matrices and proves the feasibility of their use for encrypting block data. It was found that the use of such matrices as a separate procedure for protecting block data in the theory and practice of cryptography is extremely rare. The Fibonacci polynomial matrix sequences from the 2nd to the 5th order are analyzed, according to which shortcomings in the traditional approach to forming the structure of the elements of such matrices are revealed, first of all, the number (k) of its different elements, which are Fibonacci polynomials of the not higher than (n–1) th power. It is established that for Fibonacci matrices of any order (m) of such different Fibonacci polynomials there will be only k=3 , the structure of which will depend only on the number (n) of the sequence of the Fibonacci polynomial matrix. Such a small number is not only uninformative and easily analyzable by a cryptanalyst but also vulnerable to cryptanalysis. The structure of the elements of Fibonacci polynomial matrices is specified, the number of which already depends on the order of the matrix(m) and is k=m+1 . A method for generating a sequence of refined Fibonacci polynomial matrices has been developed, which consists in using a recurrent matrix relation, according to which the next polynomial matrix is formed by multiplying the variable x sequentially by the elements of the current matrix, adding the elements of the newly generated matrix to those of the previous matrix, after which all similar terms are grouped in the formed elements. The mechanisms for forming a sequence of 8 polynomial refined Fibonacci matrices from the 2nd to the 4th order, the elements of which are Fibonacci polynomials of no higher than (m+n–2) th power, have been presented, which made it possible to analyze not only the features of their construction, but also to understand the corresponding procedures for finding their determinants and inverse matrices. It was found that the proposed structure of the elements of the nth sequence of Fibonacci polynomial matrices of the mth order has an interesting property, according to which it is possible to avoid the use of the recurrent matrix relation, and to generate the corresponding Fibonacci polynomial matrices only by the numbers (m+n–2–j) th sequence of Fibonacci polynomials, the specific values of which depend on their location in the matrix and the number of its column, namely ∀j∈[0÷(m−1)] . Software has been developed that allows generating both sequences of refined Fibonacci polynomial matrices of the mth order, and finding their determinants and inverse polynomial matrices of a similar order. An example of the application of the matrix method of encrypting block data with a refined Fibonacci polynomial matrix is given, which allows the interested reader to understand the basic principle of encrypting both the initial message and decrypting the encrypted message.

Polynomial Equations Giving a Proper Feedback Compensator for a Strictly Proper Plant

We consider the problem of determining whether two polynomial matrices can be transformed to one another by left multiplying with some nonsingular numerical matrix and right multiplying by some invertible polynomial matrix. Thus the equivalence relation arises. This equivalence relation is known as semiscalar equivalence. Large difficulties in this problem arise already for 2-by-2 matrices. In this paper the semiscalar equivalence of polynomial matrices of second order is investigated. In particular, necessary and sufficient conditions are found for two matrices of second order being semiscalarly equivalent. The main result is stated in terms of determinants of Toeplitz matrices.

The objective of this paper is to derive formulae for the generalized Drazin inverse of a block matrix in a Banach algebra \(\mathcal{A}\) under different conditions. Let \(x=\left[ \begin{array}{c@{\quad }c} a&{}b\\ c&{}d\end{array}\right] \in \mathcal{A}\) relative to the idempotent \(p\in \mathcal{A}\) and \(a\in p\mathcal{A}p\) be generalized Drazin invertible. The formulae for the generalized Drazin inverse are obtained under the more general case that the generalized Schur complement \(s=d-ca^db\) is generalized Drazin invertible, which covers the cases that \(s\) is Drazin invertible, \(s\) is group invertible, or \(s\) is equal to zero. Thus, recent results on the Drazin inverse of block matrices and block-operator matrices are extended to a more general setting.

RANK ADDITIVITY AND MATRIX POLYNOMIALS

Spaces of multiparameter rational vectors, i.e., of vectors whose components are rational functions in several variables, and polynomial bases of their subspaces are considered. The conjecture that any subspace in the space of multiparameter rational vectors possesses a “free” polynomial basis, i.e., a basis such that the associated basis multiparameter polynomial matrix has no finite regular spectrum, is disproved by an example. Some consequences of this fact are indicated. Simpler proofs of some properties of the singular spectra of basis polynomial matrices corresponding to the null-spaces of a singular polynomial matrix are presented. Bibliography: 5 titles.

We present a computationally efficient algorithm for computing the 2-D Capon spectral estimator. The implementation is based on the fact that the 2-D data covariance matrix will have a Toeplitz-Block-Toeplitz structure, with the result that the inverse covariance matrix can be expressed in closed form by using a special case of the Gohberg-Heinig formula that is a function of strictly the forward 2-D prediction matrix polynomials. Numerical simulations illustrate the clear computational gain in comparison to both the well-known classical implementation and the method recently published by Liu et al. (see IEEE Trans. Aerosp. Electron. Syst., vol.34, no.4, p.1314-19, October 1998).

The representations for the Drazin inverse of anti-triangular matrices are obtained under some conditions. Applying these representations, we give a necessary condition for a class of block matrices to have signed Drazin inverse.