Spectral properties of $\lambda$-matrices

One investigates the spectral problem for polynomial $\lambda$-matrices of the general form (regular and singular). One establishes a relation between the elements of the complete spectral structure of the $\lambda$-matrix (i.e., its elementary divisors, its Jordan chains of vectors, its minimal indices and polynomial solutions) and of the matrices which occur in its transformation to the Smith canonical form. One establishes a correspondence between the complete spectral structures of a $\lambda$-matrix and of three linear accompanying matrix pencils. One notes the possibility of reducing the solution of the spectral problem for a $\lambda$-matrix to the solution of a similar problem for its accompanying pencil. One gives a factorization of a $\lambda$-matrix which allows to represent it by any of its considered accompanying pencils or by its Kronecker canonical form.