On common unital divisors of matrix polynomials over an arbitrary field

The problem of common left divisors of matrix polynomials over an arbitrary field is investigated. A necessary and, with certain restrictions, sufficient condition is established for the existence of a common unital divisor with a prescribed characteristic polynomial of two nonsingular matrix polynomials. The approach offered in this paper has made it possible to solve this problem completely for matrix polynomials if at least one of them has pairwise relatively prime elementary divisors. In the case when the desired divisor exists, a method of constructing it is specified.