The paper considers conditions under which rank factorizations of a two-parameter polynomial matrix can be affected with the use of unimodular matrices, as in the one-parameter case. Algorithms for computing such factorizations and a minimal basis of the null space of the corresponding matrix are presented. Also an algorithm for inverting unimodular two-parameter polynomial matrices is suggested. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]
The paper continues the development of rank-factorization methods for solving certain algebraic problems for multi-parameter polynomial matrices and introduces a new rank factorization of a q-parameter polynomial m × n matrix F of full row rank (called the PG-q factorization) of the form F = PG, where $$P = \prod\limits_{k = 1}^{q - 1} {\prod\limits_{i = 1}^{n_k } {\nabla _i^{(k)} } } $$ is the greatest left divisor of F; Δ i is a regular (q-k)-parameter polynomial matrix the characteristic polynomial of which is a primitive polynomial over the ring of polynomials in q-k-1 variables, and G is a q-parameter polynomial matrix of rank m. The PG-q algorithm is suggested, and its applications to solving some problems of algebra are presented. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]
The paper continues the investigation of methods for factorizing q-parameter polynomial matrices and considers their applications to solving multiparameter problems of algebra. An extension of the AB-algorithm, suggested earlier as a method for solving spectral problems for matrix pencils of the form A - λB, to the case of q-parameter (q ≥ 1) polynomial matrices of full rank is proposed. In accordance with the AB-algorithm, a finite sequence of q-parameter polynomial matrices such that every subsequent matrix provides a basis of the null-space of polynomial solutions of its transposed predecessor is constructed. A certain rule for selecting specific basis matrices is described. Applications of the AB-algorithm to computing complete polynomials of a q-parameter polynomial matrix and exhausting them from the regular spectrum of the matrix, to constructing irreducible factorizations of rational matrices satisfying certain assumptions, and to computing “free” bases of the null-spaces of polynomial solutions of an arbitrary q-parameter polynomial matrix are considered. Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]