Monic Matrix Polynomials Research Articles

On the quasi-stability criteria of monic matrix polynomials

This paper is a continuation of a recent investigation by Zhan and Dyachenko (2021) on the Hurwitz stability of monic matrix polynomials with algebraic techniques. By improving an inertia formula for matrix polynomials with respect to the imaginary axis, we show that, under some conditions, the quasi-stability of a monic matrix polynomial can be tested via the Hermitian nonnegative definiteness of two block Hankel matrices built from its matricial Markov parameters. Moreover, for the so-called doubly monic matrix polynomials, the quasi-stability criteria can be formulated in a much simpler form. In particular, the relationship between Hurwitz stable matrix polynomials and Stieltjes positive definite matrix sequences established in Zhan and Dyachenko (2021) is included as a special case.

New bounds for the Eigenvalues of Matrix Polynomials

We employ several numerical radius inequalities to the square of the Frobenius companion matrices of monic matrix polynomials to provide new bounds for the eigenvalues of these polynomials.

On the relation between Hurwitz stability of matrix polynomials and matrix-valued Stieltjes functions

In this paper, we elaborate on the relationship between the Hurwitz stability of matrix polynomials and matrix-valued Stieltjes functions. Our strategy is that, for a monic matrix polynomial, we associate a rational matrix-valued function with its even–odd split and then check the Hurwitz stability of the matrix polynomial by testing the Stieltjes property of the related rational matrix-valued function. On the basis of this relationship, we present matrix generalizations of a classical stability criterion by Gantmacher, Chebotarev theorem, Grommer theorem and some aspects of the modified Hermite–Biehler theorem. Our work is motivated by one of the authors’ recent stability studies linked with matricial Markov parameters.

The Limit Empirical Spectral Distribution of Gaussian Monic Complex Matrix Polynomials

We define the empirical spectral distribution (ESD) of a random matrix polynomial with invertible leading coefficient, and we study it for complex n times n Gaussian monic matrix polynomials of degree k. We obtain exact formulae for the almost sure limit of the ESD in two distinct scenarios: (1) n rightarrow infty with k constant and (2) k rightarrow infty with n constant. The main tool for our approach is the replacement principle by Tao, Vu and Krishnapur. Along the way, we also develop some auxiliary results of potential independent interest: We slightly extend a result by Bürgisser and Cucker on the tail bound for the norm of the pseudoinverse of a nonzero mean matrix, and we obtain several estimates on the singular values of certain structured random matrices.

Prime decomposition of quadratic matrix polynomials

<abstract> We study the prime decomposition of a quadratic monic matrix polynomial. From the prime decomposition of a quadratic matrix polynomial, we obtain a formula of the general solution to the corresponding second-order differential equation. For a quadratic matrix polynomial with pairwise commuting coefficients, we get a sufficient condition for the existence of a prime decomposition. </abstract>

Bounds for the eigenvalues of monic matrix polynomials from numerical radius inequalities

We apply several numerical radius inequalities to the Frobenius companion matrices of monic matrix polynomials to derive new bounds for the eigenvalues of these polynomials. One of our bounds improves upon an earlier bound due to Higham and Tisseur.

On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials

This paper is a direct continuation of the author's recent investigations [4] on the non-degenerate truncated matricial Stieltjes problem. Inspired by a characterization of scalar Hurwitz polynomials via continued fractions (see e.g. Gantmacher [15]) a class of matrix Hurwitz type polynomials is introduced. It is shown that this class is intimately related with Stieltjes positive definite sequences of matrices and can be expressed in terms of the associated Stieltjes quadruple of sequences of left orthogonal matrix polynomials. It is shown that a monic matrix polynomial is a matrix Hurwitz type polynomial if and only if the sequence of its Markov parameters is a Stieltjes positive definite sequence.

An algorithm for computing the eigenvalues of block companion matrices

In this paper we propose a method for computing the roots of a monic matrix polynomial. To this end we compute the eigenvalues of the corresponding block companion matrix C. This is done by implementing the QR algorithm in such a way that it exploits the rank structure of the matrix. Because of this structure, we can represent the matrix in Givens-weight representation. A similar method as in Chandrasekaran et al. (Oper Theory Adv Appl 179:111---143, 2007), the bulge chasing, is used during the QR iteration. For practical usage, matrix C has to be brought in Hessenberg form before the QR iteration starts. During the QR iteration and the transformation to Hessenberg form, the property of the matrix being unitary plus low rank numerically deteriorates. A method to restore this property is used.

Polynomial numerical hulls of matrix polynomials, II

In this article, we study some algebraic and geometrical properties of polynomial numerical hulls of matrix polynomials and joint polynomial numerical hulls of a finite family of matrices (possibly the coefficients of a matrix polynomial). Also, we study polynomial numerical hulls of basic A-factor block circulant matrices. These are block companion matrices of particular simple monic matrix polynomials. By studying the polynomial numerical hulls of the Kronecker product of two matrices, we characterize the polynomial numerical hulls of unitary basic A-factor block circulant matrices.

The polar decomposition of block companion matrices

Let L(λ) = I n λ m + A m−1 λ m−1 + …+ A 1λ + A 0 be an n × n monic matrix polynomial, and let C L be the corresponding block companion matrix. In this note, we extend a known result on scalar polynomials to obtain a formula for the polar decomposition of C L when the matrices A 0 and Σ j=1 m−1 A jA* j are nonsingular.

On spectra of expansion graphs and matrix polynomials

An expansion graph of a directed weighted graph G 0 is obtained by replacing certain edges of G 0 by disjoint chains. The adjacency matrix of the expansion graph is a partial linearization of a monic matrix polynomial. We prove results on common properties of a monic operator polynomial and its partial linearization. The graph G 0 is connected if and only if each expansion graph of G 0 is connected; in this case we compute the index of imprimitivity of the adjacency matrix of some special expansion graphs of G 0.

On the connectedness of numerical range of matrix polynomials

An investigation on nonconnectedness of numerical range for monic matrix polynomials L( λ) is undertaking here. The distribution of eigenvalues of L( λ) to the components of numerical range and some other algebraic properties are also presented.

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.

Some results on numerical ranges and factorizations of matrix polynomials

Some new factorization theorems for monic matrix polynomials are obtained. These theorems are based on the numerical range having the number of connected components equal to the degree of the polynomial. For second degree polynomials, sufficient conditions are given for the numerical range to have two connected components.

Robust stability of linear systems described by higher-order dynamic equations

The stability radius of higher-order differential and difference systems with respect to various classes of complex affine perturbations of the coefficient matrices is studied. Different perturbation norms are considered. The aim is to derive robustness criteria that are expressed directly in terms of the original data. Previous results on robust stability of Hurwitz and Schur polynomials are extended to monic matrix polynomials. For disturbances acting via a uniform input matrix, computable formulas are obtained, whereas for perturbations with multiple input matrices, structured singular values are involved. >

Decomposable matrix polynomials

Decomposability of a monic matrix polynomial is defined as it was by Colojoara and Foias for a single matrix (operator). It is shown that decomposable polynomials have properties close to those of linear ones. They are characterized in the general case and also when they are products of linear factors.

A computer-aided method for solvents and spectral factors of matrix polynomials

A new approach for determining the complete sets of solvents and spectral factors of a monic matrix polynomial is proposed. A systematic method for determining the initial guess for the extended multidimensional Newton-Raphson method is first proposed, such that the eigenspectrum corresponding to each solvent of the matrix polynomial can be determined. With the evaluated eigenspectra, complete sets of solvents and spectral factors of a monic matrix polynomial can be obtained by utilizing the applications and the advantages of the principal- nth-root method, the matrix sign function, and the block-power method. The established algorithms can be applied in the analysis and/or design of systems described by high-degree vector differential equations and/or matrix fractions.

A theorem on the factorization of matrix polynomials

A similarity condition is developed for the factorization of monic matrix polynomials L (λ) into the forms L (λ) = L k (λ) … L 1 (λ), wihtout any restriction on the spectrum of factors L j (λ).

On minimal factorizations of rational matrix functions

The theory of factorization of rational matrix functions W(λ) = = C(λI-A)−1B + D, as presented in the book Bart-Gohberg-Kaashoek [1], is extended here to the case where D = W(∞) is not invertible, and applied to factorizations of monic matrix polynomials.

Spectral analysis of regular matrix polynomials

INTRODUCTION A spectral method for the analysis of monic matrix polynomials was developed recently in [8-10]. A similar method was developed [11-12] for non-monic, especially comonic, matrix polynomials. The present work suggests a unified approach, which extends the above methods and results to the class of all regular matrix polynomials (i.e. with non-zero determinant). The spectral method basically concerns the construction of a pair of matrices (X,T) , called a standard pair, from the spectral data of the given matrix polynomial A(I) , and is described in Sections 1-4. The inverse problem, namely the construction of A(1) from the pair (X,T] , is described in Section 5. In Section 6 we discuss a representation for A-l(1) and its application to basic linear systems of equations, and in Section 7, all factorizations of A(I) are characterized by certain restrictions on the pair (X,T) .