Michal Wojtylak

Christian Mehl, Volker Mehrmann, Michal Wojtylak

A wide class of matrix pencils connected with dissipative Hamiltonian descriptor systems is investigated. In particular, the following properties are shown: all eigenvalues are in the closed left half plane, the nonzero finite eigenvalues on the imaginary axis are semisimple, the index is at most two, and there are restrictions for the possible left and right minimal indices. For the case that the eigenvalue zero is not semisimple, a structure-preserving method is presented that perturbs the given system into a Lyapunov stable system.

Anna Kula, Michal Wojtylak, Janusz Wysoczański

Rank two parametric perturbations of operators and matrices are studied in various settings. In the finite dimensional case the formula for a characteristic polynomial is derived and the large parameter asymptotics of the spectrum is computed. The large parameter asymptotics of a rank one perturbation of singular values and condition number are discussed as well. In the operator case the formula for a rank two transformation of the spectral measure is derived and it appears to be the t-transformation of a probability measure, studied previously in the free probability context. New transformation of measures is studied and several examples are presented.

Luca Perotti, Michal Wojtylak

A representation of the Padé approximation of the $Z$-transform of a signal as a resolvent of a tridiagonal matrix $J$ is given. Several formulas for the poles, zeros and residues of the Padé approximation in terms of the matrix $J$ are proposed. Their numerical stability is tested and compared. Methods for computing forward and backward errors are presented.