The generalized Levinger transformation

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.