The generalized Levinger transformation

Abstract

In this paper, we present new results relating the numerical range of a matrix $A$ with generalized Levinger transformation $\mathcal{L}(A,α,β) = \alphaH_A +\betaS_A$, 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.

8th Workshop on "Numerical Ranges and Numerical Radii" (WONRA), University of Bremen, 2006