Constructing symmetric structure-preserving strong linearizations

Polynomials eigenvalue problems with structured matrix polynomials arise in many applications. The standard way to solve polynomial eigenvalue problems is through the classical Frobenius companion linearizations, which may not retain the structure of the matrix polynomial. Particularly, the structure of the symmetric matrix polynomials can be lost, while from the computational point of view, it is advisable to construct a linearization which preserves the symmetry structure. Recently, new families of block-Kronecker pencils have been introduced in [5]. Applying block-Kronecker pencils, we present structure-preserving strong linearizations for symmetric matrix polynomials. When the matrix polynomial has an odd degree, these linearizations are strong regardless of whether the matrix polynomial is regular or singular. Additionally, we construct structure-preserving strong linearizations for regular symmetric matrix polynomials of even degree under some simple nonsingularity conditions.