arXiv subjects

Mohit Kumar Baghel

Publications and source records attributed to Mohit Kumar Baghel.

3 recordsLinked to original sources

arXiv2022

Doubly structured mapping problems of the form $Δx=y$ and $Δ^*z=w$

For a given class of structured matrices $\mathbb S$, we find necessary and sufficient conditions on vectors $x,w\in \C^{n+m}$ and $y,z \in \C^{n}$ for which there exists $Δ=[Δ_1~Δ_2]$ with $Δ_1 \in \mathbb S$ and $Δ_2 \in \C^{n,m}$ such that $Δx=y$ and $Δ^*z=w$. We also characterize the set of all such mappings $Δ$ and provide sufficient conditions on vectors $x,y,z$, and $w$ to investigate a $Δ$ with minimal Frobenius norm. The structured classes $\mathbb S$ we consider include (skew)-Hermitian, (skew)-symmetric, pseudo(skew)-symmetric, $J$-(skew)-symmetric, pseudo(skew)-Hermitian, positive (semi)definite, and dissipative matrices. These mappings are then used in computing the structured eigenvalue/eigenpair backward errors of matrix pencils arising in optimal control.

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arXiv2021

On the non-symmetric semidefinite Procrustes problem

In this paper, we consider the non-symmetric positive semidefinite Procrustes (NSPSDP) problem: Given two matrices $X,Y \in \mathbb{R}^{n,m}$, find the matrix $A \in \mathbb{R}^{n,n}$ that minimizes the Frobenius norm of $AX-Y$ and which is such that $A+A^T$ is positive semidefinite. We generalize the semi-analytical approach for the symmetric positive semidefinite Procrustes problem, where $A$ is required to be positive semidefinite, that was proposed by Gillis and Sharma (A semi-analytical approach for the positive semidefinite Procrustes problem, Linear Algebra Appl. 540, 112-137, 2018). As for the symmetric case, we first show that the NSPSDP problem can be reduced to a smaller NSPSDP problem that always has a unique solution and where the matrix $X$ is diagonal and has full rank. Then, an efficient semi-analytical algorithm to solve the NSPSDP problem is proposed, solving the smaller and well-posed problem with a fast gradient method which guarantees a linear rate of convergence. This algorithm is also applicable to solve the complex NSPSDP problem, where $X,Y \in \mathbb{C}^{n,m}$, as we show the complex NSPSDP problem can be written as an overparametrized real NSPSDP problem. The efficiency of the proposed algorithm is illustrated on several numerical examples.

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arXiv2021

Characterization of the dissipative mappings and their application to perturbations of dissipative-Hamiltonian systems

In this paper, we find necessary and sufficient conditions to identify pairs of matrices $X$ and $Y$ for which there exists $Δ\in \mathbb C^{n,n}$ such that $Δ+Δ^*$ is positive semidefinite and $ΔX=Y$. Such a $Δ$ is called a dissipative mapping taking $X$ to $Y$. We also provide two different characterizations for the set of all dissipative mappings, and use them to characterize the unique dissipative mapping with minimal Frobenius norm. The minimal-norm dissipative mapping is then used to determine the distance to asymptotic instability for dissipative-Hamiltonian systems under general structure-preserving perturbations. We illustrate our results over some numerical examples and compare them with those of Mehl, Mehrmann and Sharma (Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations, SIAM J. Mat. Anal. Appl.\ 37 (4): 1625-1654, 2016).

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