Rafikul Alam

Perturbation theory for Hamiltonian matrices and the distance to bounded-realness

Motivated by the analysis of passive control systems, we undertake a detailed perturbation analysis of Hamiltonian matrices that have eigenvalues on the imaginary axis. We construct minimal Hamiltonian perturbations that move and coalesce eigenvalues of opposite sign characteristic to form multiple eigenvalues with mixed sign characteristics, which are then moved from the imaginary axis to specific locations in the complex plane by small Hamiltonian perturbations. We also present a numerical method to compute upper bounds for the minimal perturbations that move all eigenvalues of a given Hamiltonian matrix outside a vertical strip along the imaginary axis.

Structured Backward Errors and Pseudospectra of Structured Matrix Pencils

Structured backward perturbation analysis plays an important role in the accuracy assessment of computed eigenelements of structured eigenvalue problems. We undertake a detailed structured backward perturbation analysis of approximate eigenelements of linearly structured matrix pencils. The structures we consider include, for example, symmetric, skew-symmetric, Hermitian, skew-Hermitian, even, odd, palindromic, and Hamiltonian matrix pencils. We also analyze structured backward errors of approximate eigenvalues and structured pseudospectra of structured matrix pencils.

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.

Linearizations for Rational Matrix Functions and Rosenbrock System Polynomials

Our aim in this paper is twofold: First, for solving rational eigenproblems we introduce linearizations of rational matrix functions and propose a framework for their constructions via minimal realizations. We also introduce Fiedler-like pencils for rational matrix functions and show that the Fiedler-like pencils are linearizations of the rational matrix functions. Second, for computing zeros of a linear time-invariant system

Stability of eigenvalues and spectral decompositions under linear perturbation

\Sigma

Effect of linear perturbation on spectra of matrices

in state-space form, we introduce a trimmed structured linearization, which we refer to as a Rosenbrock linearization, of the Rosenbrock system polynomial

Generalized Fiedler Pencils for Rational Matrix Functions

\mathcal{S}(\lambda)

On sensitivity of eigenvalues and eigendecompositions of matrices

associated with

Characterization and construction of the nearest defective matrix via coalescence of pseudospectral components

\Sigma.

Pseudospectra, critical points and multiple eigenvalues of matrix polynomials

Also we introduce Fiedler-like pencils for