Speaker
Description
The dissipative Hamiltonian (dH) matrix pencils are pencils of the form $L(\lambda) = \lambda E - (J-R)Q,$ where $J^* = -J$, $E^*Q = Q^*E \geq 0$, $R^* = R \geq 0$, and $\lambda E - Q$ is regular. Matrix pencils strictly equivalent to dH pencils were characterised in [2].
In this talk, we investigate the orbit structure of dH matrix pencils, in the setting proposed by Pokrzywa [3]. In particular, we determine, in terms of the Kronecker canonical form, when the closure of an orbit of a dH pencil contains only matrix pencils that are strictly equivalent to dH pencils. We also discuss the special case $Q = I$.
Furthermore, we characterise matrix pencils that are strictly equivalent to possibly singular and possibly non-square pencils of the form $P(\lambda) = \lambda E - Q$ with $E^*Q = Q^*E \geq 0$, extending results from [1].
References
[1] C. Mehl, V. Mehrmann, M. Wojtylak, Linear algebra properties of dissipative Hamiltonian descriptor systems, SIAM J. Matrix Anal. Appl., 39 (3), 1489-1519, 2018.
[2] C. Mehl, V. Mehrmann, M. Wojtylak, Matrix pencils with coefficients that have positive semidefinite Hermitian parts, SIAM J. Matrix Anal. Appl., 43 (3), 1186-1212, 2022.
[3] A. Pokrzywa, On perturbations and the equivalence orbit of a matrix pencil, Linear Algebra Appl., 82, 99-121, 1986.