Diagonalization of quadratic matrix polynomials

Solving the quadratic eigenvalue problem is critical in several applications in control and systems theory. One alternative to solve this problem is to reduce the matrix to a diagonal form so that its eigenvalue structure can be recognized in the diagonal of the equivalent matrix. There are two major categories of diagonalizable systems. The first category concerns systems that are strictly equivalent. The second category is much wider and consists of systems for which their linearizations are strictly equivalent. Here we are concerned with methods to reduce the linearization of a quadratic matrix polynomial to a diagonal form. We give necessary and sufficient conditions for a system to have a diagonalization and we argue on two different methods to diagonalize a system (via its linearization) that one can find in the literature. Based on the results presented here, we conclude that the problem is still open.