Systematic and quite intensive studies of matrix polynomials with respect to their decomposition into factors began in the second half of the present century, and was connected mainly with problems that arise in the theory of differential equations [32], the theory of operator pencils [30,31], in applied problems of oscillating [45] and dynamic systems [1], optimal control [2], and other areas. In the papers of Lancaster, Langer, Wimmer, Dennis, Traub, Weber, Gohberg, Rodman, Markus, Mereuzzi, Malishev, Yakubovich. and others, problems on the factorizability of matrix polynomials under various constraints were' studied by methods based on the classical concepts of eigenvectors and Jordan chains. P. S. Kazimirs'kii has proposed a new approach to the study of these questions. On the basis of the fundamental concepts he introduced--the value of a polynomial matrix on a system of roots of a polynomial, the companion matrix, semiscalar equivalence of polynomial matrices--algebraic methods were developed for factoring matrix polynomials, which also find effective application in the study of other questions of the theory of polynomial matrices, in particular similarity of matrix polynomials and finite sets of numerical matrices and their reducibility to simple forms. The present paper is devoted to a survey of the basic results obtained by Kazimirs'kii in developing the theory of factorization of matrix polynomials. The discussion of the material will introduce, to the extent necessary, some modern terminology that is not in the original, in order to present certain results in the most general form.
