Monic divisors with one invariant factor of polynomial matrices

Abstract

The problem of separation of a regular factor from a matrix polynomial has been of interest to mathematicians for a long time because the solution of this problem gives a method for the determination of roots of matrix one-sided equations. Various methods of contemporary mathematics are used for the solution of this problem. In 1956, Lopatyns’kyi, in terms of systems of differential equations, gave necessary and sufficient conditions for representation of a matrix in the form of the product of regular factors with coprime determinants. Gochberg, Lancaster, and Rodman in [7] and Malyshev in [4] solve this problem by the methods of Jordan chains. In [3], Kazimirs’kyi proposes an essentially new approach to the problem of separation of a regular divisor from a matrix polynomial. This approach is based on the notions of a determining matrix and values of a matrix on a system of roots of a polynomial. In these terms, Kazimirs’kyi established necessary and sufficient conditions for the existence of these divisors and proposes a method for their determination. The obtained results hold for polynomial matrices over an algebraically closed field of characteristic zero. The specific feature of methods of solution and terms used in the representation of results do not enable one to generalize these methods for wider classes of fields. This generalization became possible after work [2] in which the Kazimirs’kyi results are represented in another form. In the present paper, we investigate polynomial matrices over an infinite field. Under certain restrictions on the canonical diagonal form of a divisor, necessary and sufficient conditions for its existence are established. It is worth noting that, after the solution of the problem of separation of a regular divisor by Kazimirs’kyi, the number of works devoted to regular divisors sharply dropped. Problems of description of nonassociative divisors of matrices [6, 10] and representation of matrices in the form of the product of several factors with preassigned characteristics [5, 8, 13] have attracted the attention of researchers. Let F be a field and let A(x) be a nonsingular (n × n) matrix over F[x] representable in the form of a matrix polynomial over the field F : A(x) = A k x k + A k−1 x k−1 +… + A 0 . The matrix A(x) is called monic if A k = E is an identity matrix of order k and regular if det A k ≠ 0. We say that the matrix A(x) is right regularizable if there exists an invertible matrix U (x) such that