Computing the invariant polynomials of a polynomial matrix. 2

The paper considers the problem of computing the invariant polynomials of a general (regular or singular) one-parameter polynomial matrix. Two new direct methods for computing invariant polynomials, based on the $\Delta W$ and $\nabla V$ rank-factorization methods, are suggested. Each of the methods may be regarded as a method for successively exhausting zeros of invariant polynomials from the matrix spectrum. Application of the methods to computing adjoint matrices for regular polynomial matrices, to finding the canonical decomposition into a product of regular matrices such that the characteristic polynomial of each of them coincides whith the corresponding invariant polynomial, and to computing matrix eigenvectors corresponding to the zeros of its invariant polynomials are considered.