arXiv subjects

Rafikul Alam

Publications and source records attributed to Rafikul Alam.

8 recordsLinked to original sources

arXiv2025

Analytic matrix descriptions with application to time-delay systems

A polynomial matrix description(PMD) of a rational matrix $G(\lambda)$ is a matrix polynomial of the form $$ \mathbf{P}(\lambda) := \left[\begin{array}{c|c} A(\lambda) & B(\lambda) \\ \hline -C(\lambda) & D(\lambda)\end{array}\right] \text{ such that } G(\lambda) = D(\lambda) + C(\lambda) A(\lambda)^{-1} B(\lambda),$$ where $A(\lambda)$ is regular and called the state matrix. PMDs have been studied extensively in the context of higher order linear time-invariant systems. An analytic matrix description(AMD) of a meromorphic matrix $ M(\lambda)$ is a holomorphic matrix (i.e., a holomorphic matrix-valued function) of the form $$ \mathbf{H}(\lambda) := \left[\begin{array}{c|c} A(\lambda) & B(\lambda) \\ \hline -C(\lambda) & D(\lambda)\end{array}\right] \text{ such that } M(\lambda) = D(\lambda) + C(\lambda) A(\lambda)^{-1} B(\lambda),$$ where $A(\lambda)$ is regular and called the state matrix. AMDs arise in the study of linear time-invariant time-delay systems (TDS). Our aim is to develop a framework for analysis of AMDs analogous to the framework for PMDs and discuss the extent to which results of PMDs can be generalized to AMDs. We show that important results (e.g., coprime matrix-fraction descriptions (MFDs), least order PMDs, equivalence of PMDs, canonical forms, characterization of PMDs by transfer functions, structural indices of zeros and poles) which hold for PMDs can be generalized to AMDs which in turn can be utilized to analyze time-delay systems.

math.SP↗

arXiv2025

A brief overview of spectral perturbation Theory

The aim of this article is to present a brief overview of spectral perturbation theory for matrices, bounded linear operators and holomorphic operator-valued functions. We focus on bounds for perturbed eigenvalues, eigenvectors and invariant subspaces and provide simplified proofs of some well known results. We present a comprehensive perturbation analysis of invariant subspaces of matrices. For bounded linear operators we discuss, among other things, the effect of analytic perturbation on the discrete eigenvalues and spectral projections. We also briefly discuss analytic spectral perturbation theory for holomorphic operator-valued functions.

math.SP↗

arXiv2025

Structure-preserving deflation of critical eigenvalues in quadratic eigenvalue problems associated with damped mass-spring systems

For a quadratic matrix polynomial associated with a damped mass-spring system there are three types of critical eigenvalues, the eigenvalues $\infty$ and $0$ and the eigenvalues on the imaginary axis. All these are on the boundary of the set of (robustly) stable eigenvalues. For numerical methods, but also for (robust) stability analysis, it is desirable to deflate such eigenvalues by projecting the matrix polynomial to a lower dimensional subspace before computing the other eigenvalues and eigenvectors. We describe structure-preserving deflation strategies that deflate these eigenvalues via a trimmed structure-preserving linearization. We employ these results for the special case of hyperbolic problems. We also analyze the effect of a (possibly low rank) parametric damping matrix on purely imaginary eigenvalues.

math.NA↗

arXiv2020

Structured strong linearizations of structured rational matrices

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational matrices have been introduced recently for computing poles, eigenvalues, eigenvectors, minimal bases and minimal indices of rational matrices. For structured rational matrices, it is desirable to construct structure-preserving linearizations so as to preserve the symmetry in the eigenvalues and poles of the rational matrices. With a view to constructing structure-preserving linearizations of structured rational matrices, we propose a family of Fiedler-like pencils and show that the family of Fiedler-like pencils is a rich source of structure-preserving strong linearizations of structured rational matrices. We construct symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, skew-Hermitian, para-Hermitian and para-skew-Hermitian strong linearizations of a rational matrix $G(λ)$ when $G(λ)$ has the same structure. Further, when $G(λ)$ is real and symmetric, we show that the transfer functions of real symmetric linearizations of $G(λ)$ preserve the Cauchy-Maslov index of $G(λ).$ We describe the recovery of eigenvectors, minimal bases and minimal indices of $G(λ)$ from those of the linearizations of $G(λ)$ and show that the recovery is operation-free.

math.NA↗

arXiv2015

Linearizations for Rosenbrock system polynomials and rational matrix functions

Our aim in this paper is two-fold: First, for computing zeros of a linear time-invariant (LTI) system $\Sigma$ 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 $\Sigma.$ 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 $\Sigma_G$ in state-space form via state-space realization, we show that the Fiedler-like pencils of the Rosenbrock system polynomial associated with $\Sigma_G$ are "linearizations" of $G(\lam)$ when the system $\Sigma_G$ is both controllable and observable.

math.NA↗

arXiv2015

Structured inverse least-squares problem for structured matrices

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.

math.NA↗

arXiv2013

Structured mapping problems for linearly structured matrices

Given an appropriate class of structured matrices S; we characterize matrices X and B for which there exists a matrix A \in S such that AX = B and determine all matrices in S mapping X to B. We also determine all matrices in S mapping X to B and having the smallest norm. We use these results to investigate structured backward errors of approximate eigenpairs and approximate invariant subspaces, and structured pseudospectra of structured matrices.

math.NA↗

arXiv2009

On backward errors of structured polynomial eigenproblems solved by structure preserving linearizations

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.

math.NA↗