Sk. Safique Ahmad

On Pseudospectra, Critical Points, and Multiple Eigenvalues of Matrix Pencils

We develop a general framework for defining and analyzing pseudospectra of matrix pencils. The framework so developed unifies various definitions of pseudospectra of matrix pencils proposed in the literature. We introduce and analyze critical points of backward errors of approximate eigenvalues of matrix pencils and show that each critical point is a multiple eigenvalue of an appropriately perturbed pencil. We show that common boundary points of components of pseudospectra of a matrix pencil are critical points. In particular, we show that a minimal critical point can be read off from the pseudospectra of matrix pencils. Hence we show that a solution of Wilkinsons problem for a matrix pencil can be read off from the pseudospectra of the matrix pencil.

Backward errors for eigenvalues and eigenvectors of Hermitian, skew-Hermitian, H-even and H-odd matrix polynomials

We discuss the perturbation analysis for eigenvalues and eigenvectors of structured homogeneous matrix polynomials with Hermitian, skew-Hermitian, H-even and H-odd structure. We construct minimal structured perturbations (structured backward errors) such that an approximate eigenvalue and eigenvector pair (finite or infinite eigenvalues) is an exact eigenvalue eigenvector pair of an appropriately perturbed structured matrix polynomial. We present various comparisons with unstructured backward errors and previous backward errors constructed for the non-homogeneous case and show that our results generalize previous results.

ON ESTIMATION OF TRANSIENT STOCHASTIC STABILITY OF LINEAR SYSTEMS

In this work, we propose a measure for estimating the transient stability/stabilization of stochastic systems modeled with linear Ito Stochastic Differential Equations. The measure leads to useful and tractable computation of the stochastic Brockett version of the transient stabilization problems. Some properties and bound estimates of the measure which we call the Stochastic Logarithmic Norm are also discussed. The usefulness of the Stochastic Logarithmic Norm is illustrated with examples of unstable equilibrium systems and a non-normal system.