A set of well-known and new rank equalities for orthogonal projectors are presented. A variety of consequences for orthogonal projectors, especially for the commutativity of two orthogonal projectors, are also derived

The maximal and minimal ranks of the quadratic matrix expression – ( – )( – ) with respect to and are presented. As applications, the maximal and minimal ranks of the Schur complement with respect to a reflexive generalized inverse of matrix and their various consequences are also considered