Regular Matrix Polynomials Research Articles

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.

Factorization of selfadjoint quadratic matrix polynomials with real spectrum

Factorization theorems, and properties of sets of eigenvectors, are established for regular selfadjoint quatratic matrix polynomials L(λ) whose leading coefficeint is indefinite or possibly singular, and for which all eigenvalues are real of definite type. The two linear factors obtained have spectra which are just the eigenvalues of L(λ) of positive and negative types, respectively.

A method for finding a common linear divisor of the matrix polynomials over an arbitrary field

Conditions are found which guarantee the existence of a common linear unital divisor with a given characteristic polynomial for the regular matrix polynomials over an arbitrary field. The result obtained is applied when finding a common solution of matrix polynomial equations.

On the factorization of matrix polynomials with commuting coefficients

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 algorithm for computing nonmonic second order multivariable systems

An efficient method for computing impulse response matrices of nonmonic multivariable second order systems is proposed. A procedure has been developed, which uses some properties of regular matrix polynomials P( s) = s 2 A 2 + sA 1 + A 0, with A 2 identically singular. An algorithm suitable for computer use is established and its applications can be effectively controlled via the newly introduced special graphical scheme. It produces simultaneously the adjoint and determinant of the matrix polynomial P( s) and avoids the inversion of any matrix. Moreover, all computations can be performed on constant matrices and coefficients, only. It is shown that substantial economy may be achieved in computation time and storage when solving large order problems, characteristic of many physical plants and engineering structures.

Common minimal multiples and divisors for rational matrix functions

We extend to rational matrix functions the theory of common multiples and common divisors of regular matrix polynomials developed in [12]. Attention is focused mainly on problems of existence, uniqueness, and construction from local data.

Solution of the partial eigenvalue problem for a regular matrix pencil

Algorithms are proposed for the solution of the partial eigenvalue problem for regular polynomial matrix pencils.

Leverrier’s Algorithm: A New Proof and Extensions

A new derivation is given of the Leverrier–Fadeev algorithm for simultaneous determination of the adjoint and determinant of the $n \times n$ characteristic matrix $\lambda I_n - A$. The proof uses an appropriate companion matrix and is of some interest in its own right. The method is extended to produce a corresponding scheme for the inverse of the polynomial matrix $\lambda ^2 I_n - \lambda A_1 - A_2 $, and indeed can be generalized for a regular polynomial matrix of arbitrary degree. The results.have application to linear control systems theory.

Regular matrix polynomials with given spectral data

Regular matrix polynomials with given spectral data

Some properties of the Riesz projector for regular matrix polynomials

Some properties of the Riesz projector for regular matrix polynomials

General theory of regular matrix polynomials and band Toeplitz operators

La theorie generale des polynomes matriciels reguliers est etendue et developpee de telle facon que les resultats soient symetriques par rapport aux donnees spectrales a gauche et a droite. On donne des applications a la factorisation canonique et a l'inversion des operateurs de Toeplitz a bande bloc

An algorithm for computing the eigenstructure of a regular matrix polynomial

We give an algorithm to compute the finite zeros of a regular matrix polynomial P(λ) [i.e. det P(λ) ≢ 0]. The approach is close to that of the algorithm in [8]. We use nonunitary elementary matrices instead of unitary matrices for the equivalence transformations, which are somewhat cheaper. In practice the danger of growth of nonunitary matrices seems to be more remote than usually supposed.

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) .

Reduction of a regular matrix polynomial to quasidiagonal form

Reduction of a regular matrix polynomial to quasidiagonal form