The generalized Levinger transformation

In this paper, we present new results relating the numerical range of a matrix A with the generalized Levinger transformation L ( A , α , β ) = α H A + β S A , where H A and S A , are, respectively the Hermitian and skew-Hermitian parts of A . Using these results, we then derive expressions for eigenvalues and eigenvectors of the perturbed matrix A + L ( E , α , β ) , for a fixed matrix E and α , β are real parameters.