arXiv subjects

M Hariprasad

Publications and source records attributed to M Hariprasad.

2 recordsLinked to original sources

arXiv2020

On the perturbations of well separated matrices

A matrix is well separated if all its Gershgorin circles are away from the unit circle and they are separated from each other. In this article, the region of relative errors in the eigenvalues is obtained as a quadratic oval for non diagonal perturbation of well seperated matrices. Thus giving a computable relative error bound in terms of Gershgorin circle parameters. When the separation is $O(n)$ and the matrix is positive definite, an interlacing theorem for the eigenvalues under perturbation is presented. Further when the separation is $O(n^2 )$, condition number of the eigenvector matrix is upper bounded to obtain the region of perturbed eigenvalue. Numerical results show the relation between diagonal entries and the magnitude of the eigenvector entries even when the matrix is not so well separated. We exploit this trend in estimating the Perron vector using power method.

math.GM↗

arXiv2019

Recursive eigen extrusion: Expanding eigenbasis conjecture

Consider $n$ linearly independent vectors in $\mathbb{C}^n$ which form columns of a matrix $A$. The recursive evaluation of eigen directions (normalized eigenvectors) of $A$ is the solution of an eigenvalue problem of the form $A_iX_i=X_i\Lambda_i$ with $i=0,1,2 \dots$; and here $\Lambda_i$ is the diagonal matrix of eigenvalues and columns of $X_i$ are the eigenvectors. Note that $A_{i+1}=\phi(X_i)$ where $\phi$ normalizes all eigenvectors to unit $\mathcal{L}_2$ norm such that all diagonal elements $[\phi(X)^\dagger\phi(X)]_{jj}=1$. It is to be proven that for any matrix $A_o$ and $n \leq 7$, the limiting set of matrices $A_i$ with $i \to \infty$ is the set of unitary matrices $U(n)$ with $X_i^\dagger X_i \to I$. Interestingly, this problem also represents a recursive map that maximizes some average distance among a set of $n$ points on the unit $n$-sphere. We first formally pose this conjecture, present extensive numerical results highlighting it, and prove it for special cases.

math.GM↗