The generalized Levinger transformation

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