Equalities for orthogonal projectors and their operations (Q539186)

A complex square matrix \(A\) is called an orthogonal projector if \(A^{2} = A = A^{*}\), where \(A^{*}\) denotes the conjugate transpose of \(A\). The author gives a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. He first collects some well-known rank formulas for orthogonal projectors and their operations, and then gives various new rank formulas for matrix expressions composed by orthogonal projectors. As applications, some necessary and sufficient conditions are given for various equalities for orthogonal projectors and their operations to hold.