On the spectra of some combinations of two generalized quadratic matrices

Let A and B be two generalized quadratic matrices with respect to idempotent matrices P and Q, respectively, such that (A - alpha P)(A - beta P) = 0, AP = PA = A, 113 y (2)(B - gamma Q) (B - delta Q) = 0, BQ = QB - B PQ - QP, AB not equal BA, and (A + B)(alpha beta P - gamma delta Q) - (alpha beta P - gamma delta Q)(A + B) with alpha, beta, gamma, delta is an element of C. Let A + B be diagonalizable. The relations between the spectrum of the matrix A + B and the spectra of some matrices produced from A and B are considered. Moreover, some results on the spectrum of the matrix A + B are obtained when A B is not diagonalizable. Finally, some results and examples illustrating the applications of the results in the work are given. (C) 2015 Elsevier Inc. All rights reserved.