Filters and Structure Preserving Transformations for Quadratic Systems

The spectral equivalence of quadratic systems can be characterized using matrix polynomials called spectral filters. These filters were characterized in [8] for quadratic systems with nonsingular leading coefficient in terms of structure preserving transformations (SPTs) of three types of linearizations. This paper extends the results of [8] to quadratic systems with singular leading coefficients, using a new type of linearization called the Rblock-symmetric linearization. It is then shown that, for any two spectrally equivalent quadratic systems and any of their linearizations, there is a homeomorphism between their linear spectral filters and the set of SPTs of the chosen linearization. In addition, spectral filters for spectrally equivalent quadratic systems must satisfy a Bezout identity. With a view to the decoupling problem of quadratic systems using filters, we identify matrix polynomials of small degree that realize the Bezout identity for the given spectrally equivalent quadratic systems and their filters.