To solving the eigenvalue problem for polynomial matrices of general form

The paper considers the eigenvalue problem for a polynomial $m\times n$ matrix $F(\mu)$ of rank $\rho$. Algorithms allowing one to reduce this problem to the generalized matrix eigenvalue problem are suggested. The algorithms are based on combining rank factorization methods and the method of hereditary pencils. Methods for exhausting subspaces of polynomial solutions of zero index from the matrix null-spaces and for isolating the regular kernel from $F(\mu)$, with the subsequent linearization, are proposed.