Factorizations Of Rational Matrix Functions Research Articles

On a case of -linear conjugation problem on the unit circle

The R -linear conjugation problem on the unit circle with rational coefficients is solved. The problem is equivalently reduced to the vector–matrix C -linear conjugation problem for which the canonical matrix is effectively constructed. The applied method is based on the two-fold application of G.N.Chebotarev's approach to the factorization of rational matrix functions.

On the Exact Spectral Factorization of Rational Matrix Functions with Applications to Paraunitary Filter Banks

On the Exact Spectral Factorization of Rational Matrix Functions with Applications to Paraunitary Filter Banks

Factorization of rational matrix functions and difference equations

Abstract In the beginning of the twentieth century, Plemelj introduced the notion of factorization of matrix functions. The matrix factorization finds applications in many fields such as in the diffraction theory, in the theory of differential equations and in the theory of singular integral operators. However, the explicit formulas for the factors of the factorization are known only in a few classes of matrices. In the present paper we consider a new approach to obtain the factorization of a rational matrix function, relative to the unit circle. The constructed method is based on the relation between the general solution of a homogeneous Riemann-Hilbert problem and a solution of a linear system of difference equations with constant coefficients.

Canonical factorization of rational matrix functions. A note on a paper by P. Dewilde

A left canonical factorization theorem for rational matrix functions relative to the unit circle is presented. The result is a time invariant version of a recent strict LU factorization theorem for certain semi-separable operators, due to Dewilde (2012) [6]. Explicit formulas for the factors are also given. The theorem is proved, first by using the state space method of Dewilde (2012) [6], and next by using an operator theory approach. In both cases the main part of the proof concerns rational matrix functions that are unitary on the unit circle.

The Non–symmetric Discrete Algebraic Riccati Equation and Canonical Factorization of Rational Matrix Functions on the Unit Circle

Canonical factorization of a rational matrix function on the unit circle is described explicitly in terms of a stabilizing solution of a discrete algebraic Riccati equation using a special state space representation of the symbol. The corresponding Riccati difference equation is also discussed.

Solving a Structured Quadratic Eigenvalue Problem by a Structure-Preserving Doubling Algorithm

In studying the vibration of fast trains, we encounter a palindromic quadratic eigenvalue problem (QEP) $(\lambda^{2}A^{T}+\lambda Q+A)z=0$, where $A,Q\in\mathbb{C}^{n\times n}$ and $Q^{T}=Q$. Moreover, the matrix Q is block tridiagonal and block Toeplitz, and the matrix A has only one nonzero block in the upper-right corner. So most of the eigenvalues of the QEP are zero or infinity. In a linearization approach, one typically starts with deflating these known eigenvalues for the sake of efficiency. However, this initial deflation process involves the inverses of two potentially ill-conditioned matrices. As a result, large error might be introduced into the data for the reduced problem. In this paper we propose using the solvent approach directly on the original QEP, without any deflation process. We apply a structure-preserving doubling algorithm to compute the stabilizing solution of the matrix equation $X+A^{T}X^{-1}A=Q$, whose existence is guaranteed by a result on the Wiener–Hopf factorization of rational matrix functions associated with semi-infinite block Toeplitz matrices and a generalization of Bendixson's theorem to bounded linear operators on Hilbert spaces. The doubling algorithm is shown to be well defined and quadratically convergent. The complexity of the doubling algorithm is drastically reduced by using the Sherman–Morrison–Woodbury formula and the special structures of the problem. Once the stabilizing solution is obtained, all nonzero finite eigenvalues of the QEP can be found efficiently and with the automatic reciprocal relationship, while the known eigenvalues at zero or infinity remain intact.

Ranges of Sylvester maps and a minimal rank problem

It is proved that the range of a Sylvester map defined by two matrices of sizes p × p and q × q, respectively, plus matrices whose ranks are bounded above, cover all p × q matrices. The best possible upper bound on the ranks is found in many cases. An application is made to a minimal rank problem that is motivated by the theory of minimal factorizations of rational matrix functions.

Factorizations of rational matrix functions with application to discrete isomonodromic transformations and difference Painlevé equations

We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this space is given by a mix of residue eigenvectors of the matrix and its inverse. Our approach is motivated by the theory of discrete isomonodromic transformations and their relationship with difference Painlevé equations. In particular, in these coordinates, basic isomonodromic transformations take the form of the discrete Euler–Lagrange equations. Secondly we show that dPV equations, previously obtained in this context by D Arinkin and A Borodin, can be understood as simple relationships between the residues of such matrices and their inverses.

Quasicomplete factorization and the two machine flow shop problem

A connection is made between the Two Machine Flow Shop Problem (2MFSP) from job scheduling theory and the issue of quasicomplete factorization of rational matrix functions. A quasicomplete factorization is a factorization into elementary (i.e., degree one) factors such that the number of factors is minimal. For a companion based matrix function W, the number of factors in a quasicomplete factorization of W is related in a simple way to the minimum makespan of an instance J of 2MFSP which can be associated to W. As a consequence of this result, variants of the 2MFSP and other types of factorization can be related too.

On self-adjoint matrix polynomials with constant signature

This paper mainly concerns the class of self-adjoint matrix polynomials with constant signature. We define the order of neutrality for a regular self-adjoint matrix polynomial L( λ). Suppose that the leading coefficient of L( λ) is nonsingular; then it is proved that L( λ) is of constant signature if and only if nl is even and the order of neutrality of L( λ) is equal to nl 2 , where n is the degree l is the size of the polynomial L( λ). A similar result is obtained for regular self-adjoint matrix polynomials. We give an answer to an open problem concerning symmetric factorizations of self-adjoint matrix polynomials of constant signature. Let L( λ) be a self-adjoint matrix polynomial of even degree and constant signature and with a nonsingular leading coefficient A n . If all the elementary divisors of L( λ) are linear, then L(λ) = [M( \\ ̄ gl)]∗A nM(λ) , with M( λ) a monic matrix polynomial. In particular, this shows that generically a self-adjoint matrix polynomial of even degree and constant signature admits such a factorization. The proof of this factorization result is based on a result concerning invariant maximal neutral subspaces for a matrix which is self-adjoint in an indefinite inner product. The latter result also applies to other situations, e.g., to factorizations of rational matrix functions with constant signature and to the existence of hermitian solutions for a class of algebraic Riccati equations.

Quasicomplete factorizations of rational matrix functions

It is shown that within the class ofn×n rational matrix functions which are analytic at infinity with valueW(∞)=In, any rational matrix functionW is the productW=W1...Wp of rational matrix functionsW1,...,Wp of McMillan degree one. Furthermore, such a factorization can be established with a number of factors not exceeding 2σ(W)−1, where σ(W) denotes the McMillan degree ofW.

On canonical factorization of rational matrix functions

This paper concerns the problem of canonical factorization of a rational matrix functionW(ψ) which is analytic but may benot invertible at infinity. The factors are obtained explicitly in terms of the realization of the original matrix function. The cases of symmetric factorization for selfadjoint and positive rational matrix functions are considered separately.

Left and right factorizations of rational matrix functions

The right partial indices of the symbol are described in terms of realizations of factors of the left Wiener-Hopf canonical factorization of the same symbol. The dual results are also stated. Application to Wiener-Hopf equations is considered.

Minimal-degree coprime factorizations of rational matrix functions

We describe coprime factorizations of rational matrix functions that have the degree of nonminimality as small as possible. The description is based on state-space representations. The recently developed theory of common minimal divisors of rational matrix functions serves as a main tool in the proofs.

Local minimal factorizations of rational matrix functions in terms of null and pole data: Formulas for factors

It is known that local minimal factorizations of a rational matrix function can be described in terms of local null and pole data (expressed in the form of left null-pole triples and their corestrictions) of this function. In this paper we give formulas for the factors in a local minimal factorization that corresponds to a given corestriction of the left null-pole triple.

On pseudo-canonical factorization of rational matrix functions

In this note we show that a rational matrix function W(λ) of the form W( λ) = I + U( λ), where U( λ) is strictly proper and has contractive values for all real λ, admits both left and right pseudocanonical factorization. Formulas for the factors are given explicitly in terms of a realization of U(λ).

Minimal factorization of rational matrix functions

The theorem on minimal factorization of rational matrix functions without a pole or zero at infinity is extended to arbitrary rational matrix functions. To obtain this generalization, the concept of a centered realization for possibly nonproper rational matrix function is developed. A centered realization involves a single state-space operator and has most properties of a usual state-space realization.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Laurent interpolation for rational matrix functions and a local factorization principle

A regular rational matrix function is constructed when a finite part of the Laurent series of the function and of its inverse are given at a finite number of points (provided an obvious necessary condition is satisfied). The construction is such that the function will have at most one extra pole (or zero), possibly of high multiplicity. This construction is further applied to develop local principles for factorizations of rational matrix functions.

On symmetric factorizations of rational matrix functions

In this paper symmetric factorization of selfadjoint rational matrix functions with constant signature on the extended real line is studied. The concepts of constant null and pole signature are introduced and studied from several points of view. It is shown that a selfadjoint rational matrix function W(λ) with constant null and pole signature admits a factorization which is minimal everywhere in the extended complex plane with the possible exception of one pre-selected real point.

Simultaneous reduction to companion and triangular forms of sets of matrices

The following three problems, are solved. When does a set of companion matrices admit simultaneous reduction to upper (lower) triangular forms? When do two sets of companion matriees admit simultaneous reduction to complementary triangular forms? When does a set of matrices admit simultaneous reduction to companion forms? The results concerning the first two problems involve (combinatorial} conditions on eigenvalues. The question of how to make these results coordinate free leads to the third problem. Its solution is phrased in terms of cyclic vectors. Simultaneous reduction to block companion forms is discussed too. The problem of complementary triangularization is related to that of complete factorization of rational matrix functions from systems theory.