For polynomial matrices of full rank, including matrices of the form $A-\lambda I$ and $A-\lambda B$, numerical methods for solving the following problems are suggested: find the divisors of a polynomial matrix whose spectra coincide with the roots of known divisors of its characteristic polynomial; compute the greatest common divisor of a sequence of polynomial matrices; solve the inverse eigenvalue problem for a polynomial matrix. The methods proposed are based on the $\Delta W$ and $\Delta V$ factorizations of polynomial matrices. Applications of these methods to the solution of certain algebraic problems are considered.