Monic Matrix Polynomials Research Articles

On connected components in the set of divisors of a monic matrix polynomial

Let A (λ) be a monic matrix polynomial. The topological properties (in particular, connectedness) of the set of all right monic divisors of A (λ) with the same determinant are investigated. It turns out that such set of divisors is not connected in general. Sufficient conditions (in terms of the elementary divisors of A (λ)) for its connectedness are provided.

On the spectral structure of monic matrix polynomials and the extension problem

We consider the following problem: find a monic matrix polynomial, given its companion matrix on a fixed invariant subspace, and given also the Jordan structure of this matrix on some complimentary invariant subspace. A detailed investigation is presented for the case when the additional Jordan matrix has only one point of spectrum.

Stable factorizations of monic matrix polynomials and stable invariant subspaces

The stable factorizations of a monic matrix polynomial are characterized in terms of spectral properties. Proofs are based on the divisibility theory developed by I. Gohberg, P. Lancaster and L. Rodman. A large part of the paper is devoted to a detailed analysis of stable invariant subspaces of a matrix. The results are also used to describe all stable solutions of the operator Riccati equation.

Spectral analysis of matrix polynomials— I. canonical forms and divisors

The Jordan normal form for complex matrices is extended to admit “canonical triples” of matrices for monic matrix polynomials and to “standard triples” of operators for monic operator polynomials on finite dimensional linear spaces. These ideas lead to the formulation of canonical, or standard, forms for such polynomials. The inverse problem is also investigated: When do triples of matrices (operators) determine a monic matrix (operator) polynomial for which the triple is canonical (standard)? The canonical and standard forms are very well suited to the study of division and multiplication processes. This is carried out in detail with special emphasis on the question of when matrix (or operator) polynomials have nontrivial divisors which are polynomials of the same kind.