Rafikul Alam

Bibhas Adhikari, Rafikul Alam

Given a pair of matrices X and B and an appropriate class of structured matrices S, we provide a complete solution of the structured inverse least-squares problem $min_{A\in_S} \|AX-B\|_F$. Indeed, we determine all solutions of the structured inverse least squares problem as well as those solutions which have the smallest norm. We show that there are infinitely many smallest norm solutions of the least squares problem for the spectral norm whereas the smallest norm solution is unique for the Frobenius norm.

Rafikul Alam, Namita Behera

Our aim in this paper is two-fold: First, for computing zeros of a linear time-invariant (LTI) system $Σ$ in {\em state-space form}, we introduce a "trimmed structured linearization", which we refer to as {\em Rosenbrock linearization}, of the Rosenbrock system polynomial $\mathcal{S}(\lam)$ associated with $Σ.$ We also introduce Fiedler-like matrices for $\mathcal{S}(\lam)$ and describe constructions of Fiedler-like pencils for $\mathcal{S}(\lam).$ We show that the Fiedler-like pencils of $\mathcal{S}(\lam)$ are Rosenbrock linearizations of the system polynomial $\mathcal{S}(\lam).$ Second, with a view to developing a direct method for solving rational eigenproblems, we introduce "linearization" of a rational matrix function. We describe a state-space framework for converting a rational matrix function $G(\lam)$ to an "equivalent" matrix pencil $\mathbb{L}(\lam)$ of smallest dimension such that $G(\lam)$ and $\mathbb{L}(\lam)$ have the same "eigenstructure" and we refer to such a pencil $\mathbb{L}(\lam)$ as a "linearization" of $G(\lam).$ Indeed, by treating $G(\lam)$ as the transfer function of an LTI system $Σ_G$ in state-space form via state-space realization, we show that the Fiedler-like pencils of the Rosenbrock system polynomial associated with $Σ_G$ are "linearizations" of $G(\lam)$ when the system $Σ_G$ is both controllable and observable.

Bibhas Adhikari, Rafikul Alam

First, we derive explicit computable expressions of structured backward errors of approximate eigenelements of structured matrix polynomials including symmetric, skew-symmetric, Hermitian, skew-Hermitian, even and odd polynomials. We also determine minimal structured perturbations for which approximate eigenelements are exact eigenelements of the perturbed polynomials. Next, we analyze the effect of structure preserving linearizations of structured matrix polynomials on the structured backward errors of approximate eigenelements. We identify structure preserving linearizations which have almost no adverse effect on the structured backward errors of approximate eigenelements of the polynomials. Finally, we analyze structured pseudospectra of a structured matrix polynomial and establish a partial equality between unstructured and structured pseudospectra.