Perturbation Theory of Transfer Function Matrices

Invertible bases and root vectors for analytic matrix-valued functions

“…A so-called maximal set of root polynomials is a useful tool as it provides all the local positive partial multiplicities at λ: see [5,23] as well as [25] for an extension to rational matrices. The idea of root vectors has proved useful for theoretical analysis [5,6,23,25] as well as to analyze conditioning [20,24] and to design practical algorithms for singular eigenvalue problems [14,17]. However, a careful inspection of the arguments in [5,23,25] shows that to define ker λ P (x) it suffices in fact to start from a, not necessarily minimal, basis satisfying properties (a) in Forney's Main Theorem 2.…”