Parameter depending state space descriptions of index-2-matrix polynomials Original Research Article

To a quadratic matrix polynomial P with coefficients in image , which originated from an electrical network and depending on a parameter vector image , a matrix A(q) and a parameter set Ω are assigned such that for all qset membership, variantΩ the eigenvalues of A(q) coincide with the zeros of detP(.;q). To find A(q) and the corresponding parameter set Ω two algorithms are proposed where the first one is similar to the Algorithm 3.6 of Van Dooren [Linear Algebra Appl. 27 (1979) 103]. The reason to compute A(q) parameter depending is given by the desire to apply the matrix perturbation theory of Stewart and Sun [Matrix Perturbation Theory (Academic Press, 1990)] to study the influence of q on the zeros of the determinant of P. The assumption that P is derived from the Laplace transform of a DAE system describing an electrical network implies that its coefficients admit representations containing sums of the form ∑vkqkwkT, where vk and wk are the unit vectors or the nonvanishing difference of two such vectors. This circumstance is decisive for the efficiency of our two algorithms.