The generalized Levinger transformation

TL;DR

This paper introduces new theoretical results connecting the numerical range of matrices with a generalized Levinger transformation, providing formulas for eigenvalues and eigenvectors of perturbed matrices involving Hermitian and skew-Hermitian parts.

Contribution

It extends the understanding of matrix transformations by deriving explicit eigenvalue and eigenvector expressions for a generalized Levinger transformation.

Findings

Derived expressions for eigenvalues of perturbed matrices.

Established relationships between numerical range and the transformation.

Provided formulas involving Hermitian and skew-Hermitian parts.

Abstract

In this paper, we present new results relating the numerical range of a matrix $A$ with generalized Levinger transformation $\mathcal{L}(A,\alpha,\beta) = \alphaH_A +\betaS_A$, where $H_A$ and $S_A$, are respectively the Hermitian and skew-hermitian parts of $A$. Using these results, we derive expressions for eigenvalues and eigenvectors of the perturbed matrix $A + \mathcal{L}(E,\alpha,\beta)$, for a fixed matrix $E$ and $\alpha, \beta$ are real parameters.