The spectra of some combinations of idempotent and involutive matrices

Key Words: Idempotent matrix, involutive matrix, spectrum, linear combination, diagonalization.It has been pointed out the concept of the spectrum and presented a short literature information related to idempotent and involutive matrices in the first chapter. Some fundamental concepts and properties have been given in the second chapter, and a study avaliable from the literature, which will guide for the rest of this study, has been examined in the third chapter.Let the linear combination matrix aP+bQ be diagonalizable, where P , Q are n×n complex matrices and a, b are nonzero complex numbers. It has been established some relations between the spectrum of the linear combination matrix and the spectra of some matrices derived from the matrices P and Q with P^2=P and Q^2=I in the fourth chapter. In case P^2=I and Q^2=I, some relations between the spectrum of the same linear combination matrix and again the spectra of some matrices produced from the matrices P and Q have been given in the fifth chapter. The last chapter consists of discussion and proposals.