Applications of CS decomposition in linear combinations of two orthogonal projectors

We study the spectrum and the rank of a linear combination of two orthogonal projectors. We characterize when this linear combination is EP, diagonalizable, idempotent, tripotent, involutive, nilpotent, generalized projector, and hypergeneralized projector. Also we derive the Moore–Penrose inverse of a linear combination of two orthogonal projectors in a particular case. The main tool used here is the CS decomposition.