On factorization of matrix polynomials

We study the decomposability of a regular matrix polynomial A(x)=A0x3+A1xs−7+.+As with commuting coefficients into factors under the assumption that some coefficient At has one of the following properties: At has only one elementary divisor; all the characteristic roots of the matrix At have multiplicity at most two.

The paper studies factorizations $a ( \lambda ) = b ( \lambda ) \cdot c ( \lambda )$ of $n \times n$ matrix polynomials, with emphasis on linear $b ( \lambda )$. It is assumed that $\det a ( \lambda )$ factorizes into scalar polynomials of degree 1. The tool is the concept of Keldysh chains (generalized Jordan chains). In Theorem 1 the linear independence of eigenvectors of a matrix is generalized to Keldysh chains.It is known that there is an $nm \times nm$ matrix polynomial $Kt ( m,a ( \lambda ) )$ such that the Keldysh chains of length $\leqq m$ with respect to an “eigenvalue” $\omega ( \det a ( \omega ) = 0 )$ may be identified with the nonzero vectors in the nullspace of $Kt ( m,a ( \omega ) )$. The main technique of the paper is to exploit the multiplicative property $Kt ( m,a ( \lambda ) ) = Kt ( m,b( \lambda ) ) \cdot Kt ( m,c ( \lambda ) )$ together with the mentioned generalization. Improved factorization results are obtained when det $a( \lambda )$ has more than $( n - 1 ) \cdot \deg a ( \lambd...

New set-theoretical solutions to the Yang—Baxter Relation are constructed. These solutions arise from the decompositions ``in different order'' of matrix polynomials and θ-functions. We also construct a ``local action of the symmetric group'' in these cases, generalizations of the action of the symmetric group SN given by the set-theoretical solution.

Matrix polynomials of the form $P(\lambda)=I\lambda^2+A_1\lambda+A_0$ (where A0 and A1 are n × n Hermitian matrices and $\lambda$ is a complex variable) arise in many applications. The numerical range of such a polynomial is \[ W(P) = \{\lambda \in \mathbb{C} : x^*P(\lambda)x = 0 \mbox{ for some nonzero } x \in \mathbb{C}^n \} \] and it always contains the spectrum of $P(\lambda)$, i.e., the set of zeros of $\textup{det}P(\lambda)$. Properties of the numerical range are developed in detail, taking advantage of the close connection between W(P) and the classical numerical range (field of values) of the (general) complex matrix A:=A0+iA1. Eigenvalues and nondifferentiable points on the boundary are examined and a procedure for the numerical determination of W(P) is presented and used for several illustrations. Some extensions of the theory to more general polynomials $P(\lambda)$ are also discussed, as well as special cases describing vibrating systems.

In this paper, we study the stability of matrix polynomials under structured perturbations of their coefficients. More precisely, we consider a family of matrix polynomials \[ P_u(λ)=A_d(u)λ^d+A_{d-1}(u)λ^{d-1}+\cdots+A_0(u), \] whose matrix coefficients depend continuously and semialgebraically on a parameter vector $u\in\mathbb{C}^p$. Assuming that the matrix polynomial is monic, we show that the spectrum, the $\varepsilon$-pseudospectrum, the numerical range, and the joint numerical range associated with $P_u(λ)$ define set-valued maps that are Hölder continuous with respect to the parameter $u$. Moreover, the parameter space $\mathbb{C}^p$ can be decomposed into a finite union of analytic semialgebraic submanifolds such that, on each submanifold, the eigenvalues and the Jordan pairs of $P_u(λ)$ depend analytically on $u$. We also note that most of the results remain valid if the monicity assumption is replaced by the local nonsingularity of the leading coefficient matrix $A_d(u)$. However, the monic setting is adopted throughout the paper in order to simplify the exposition and to avoid additional technical assumptions, which are required in particular for results concerning numerical ranges.

The numerical range of a selfadjoint matrix polynomial \( L\left( \lambda \right) = \sum\nolimits_j^\ell { = 0{\lambda ^j}{A_j}} \) is the set of points μ∈ℂ for which x* L(μ)x = 0 for some nonzero vector x. As for the classical eigenvalue problem (when L(λ) = λ I − A), the spectrum of L(λ) is contained in its numerical range. Properties of the numerical range are investigated with special emphasis on the cases when L(λ) has only real spectrum (and, possibly, the point at infinity) and when the coefficients of the matrix polynomial are real symmetric matrices.

In this paper, it is proven that the problem of the nonlinear factorization of a nonmonic matrix polynomial, $L( \lambda ) = L_1 ( \lambda ) L_2 ( \lambda ) \cdots L_k ( \lambda )$, where $L_2 ( \lambda ), \cdots ,L_k ( \lambda )$ are regular, is related to the strict equivalence between two appropriate pencils. These pencils are then obtained from the comonic companion matrices of matrix polynomials $M ( \lambda ),M_1 ( \lambda ), \cdots ,M_k ( \lambda )$ associated with $L( \lambda ),L_1 ( \lambda ), \cdots ,L_k ( \lambda )$.

A soft-decision Reed-Solomon decoding algorithm has been recently proposed by Koetter and Vardy, which provides significant coding gains by fully utilizing channel output reliability information. The algorithm converts this reliability information into a set of interpolation points and their multiplicities, and consists of three day steps: multiplicity matrix calculation, soft interpolation, and polynomial factorization. Large coding gains have been observed only for low-rate codes in additive white Gaussian noise channels, and with very high decoding complexity. Here we present a new suboptimal algorithm for computing the multiplicity matrix, discuss architectures for soft-decision decoders and evaluate the performance of the soft-decision algorithm on magnetic recording channels in the presence of erasures.

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.

Factorizations of Hermitian block Hankel matrices

We present a novel Newton method for canonical Wiener–Hopf and spectral factorization of matrix polynomials. The initial vector results from solving a block Toeplitz‐like system, and the Jacobi matrix governing the Newton iteration has nice structural and numerical properties. The local quadratic convergence of the method is proved and was tested numerically. For scalar polynomials of degree 20000, a superfast version of the method implemented on a laptop typically reqired about half a minute to produce an initial vector and then performed the Newton iteration within one second. In the matrix case, the method worked reproachless on a laptop with 8 Gigabyte RAM if the degree of the polynomial times the squared matrix dimension did not exceed 20000. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

We are revisiting the problem of solving a discrete nonlinear Schrödinger equation by the inverse scattering transform method, by use of the recently developed ExactMPF package within MAPLE Software. ExactMPF allows for an exact Wiener–Hopf factorization of matrix polynomials regardless of the partial indices of the matrix. The package can be widely used in various problems, where Wiener–Hopf factorization as one of the effective mathematical tools is required, as its code has already been disclosed. The analysis presented in this paper contains not only numerical examples of its use, but is also supported by appropriate and accurate a priori estimations. The procedure itself guarantees that the ExactMPF package produces all computations arithmetically exactly , and a detailed numerical analysis of various aspects of the computational algorithm and approximation strategies is provided in the case of a finite initial impulse.

: Several fast algebraic methods have been developed for the analysis and synthesis of multivariable control systems. Simple and quick methods based on the block-pulse functions and the matrix continued fractions have been derived for rapid model reduction, system identification, models conversion, rational matrix realization, and design of multivariable control systems. Also, computer-aided methods have been developed for determining the solvents, spectral factors of matrix polynomials and for evaluating the block partial fraction expansions of rational matrices. Furthermore, a quick algorithm based on the matrix sign function has been developed for rapid solution of matrix Riccati equation. The algorithm has successfully been applied to develop a fast suboptimal state-space self-tuner for linear stochastic multivariable systems.

On the connectedness of numerical range of matrix polynomials

In recent years, the tropical polynomial factorization problem, the tropical matrix decomposition problem, and the tropical multivariate quadratic equation solving problem have been proved to be NP-hard. Some asymmetric cryptographic systems based on tropical semirings have been proposed, but most of them are insecure and have been successfully attacked. In this paper, a new key exchange protocol and a new encryption protocol are proposed based on the difficulty of finding the multiple exponentiation problem of the tropical Jones matrices. The analysis results indicate that our protocol can resist various existing attacks. The complexity of attacking an MEP by adversaries is raised due to the larger number of combinations in the tropical Jones matrices compared to regular matrix polynomials. Furthermore, the index semiring is the non-negative integer cyclic matrix semiring, leading to a higher efficiency in key generation.