The spectrum of matrices depending on two idempotents

Let PP and QQ be two complex matrices satisfying P2=PP2=P and Q2=QQ2=Q. For a,ba,b nonzero complex numbers such that aP+bQaP+bQ is diagonalizable, we relate the spectrum of aP+bQaP+bQ to the spectra of P−QP−Q, PQPQ, PQPPQP and PQ−QPPQ−QP.