The polynomial eigenvalue problems, associated with matrix polynomials, plays a significant role in many applications. In the most widely used approach for solving this problem, there are infinitely many linearizations methods. This paper contains a compact review of the notion of linearization of matrix polynomials. The considerations about conditioning and backward error of the linearization methods are discussed. Finally, for some matrix polynomials with sequential matrix coefficients equal to zero matrix, the idea of linearization is used to produce some test generalized eigenvalue problem with one nonsingular matrix and one singular low rank matrix. This latter problem is necessary to study the behavior and performance of one of our earlier work that concerns an order reduction method for generalized eigenvalue problem.
