We discuss general and structured matrix polynomials which may be singular and may have eigenvalues at infinity. We discuss several real industrial applications ranging from acoustic field computations to optimal control problems.
We discuss linearization and first order formulations and their relationship to the corresponding techniques used in the treatment of systems of higher order differential equations.
In order to deal with structure preservation, we derive condensed/canonical forms that allow (partial) deflation of critical eigenvalues and the singular structure of the matrix polynomial. The remaining reduced order staircase form leads to new types of linearizations which determine the finite eigenvalues and corresponding eigenvectors.
Based on these new linearization techniques we discuss new structure preserving eigenvalue methods and present several real world numerical examples.