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−αP)(A−βP)=0,(A−αP)(A−βP)=0,AP=PA=A,AP=PA=A,(B−γQ)(B−δQ)=0,(B−γQ)(B−δQ)=0,BQ=QB=B,BQ=QB=B,PQ=QP,PQ=QP,AB ≠ BA  , and (A+B)(αβP−γδQ)=(αβP−γδQ)(A+B)(A+B)(αβP−γδQ)=(αβP−γδQ)(A+B) with α,β,γ,δ∈Cα,β,γ,δ∈C. Let A+BA+B be diagonalizable. The relations between the spectrum of the matrix A+BA+B and the spectra of some matrices produced from A and B   are considered. Moreover, some results on the spectrum of the matrix A+BA+B are obtained when A+BA+B is not diagonalizable. Finally, some results and examples illustrating the applications of the results in the work are given.