A Study on Rank Equalities For Idempotent Matrices: Investigating Properties and Applications of Idempotent Matrices and Orthogonal Projectors

Abstract

A square matrix A   over the complexfield C is said to be idempotent if A2=A; said to bean orthogonal projector if it is both idempotent and Hermitian, i.e., A2=A=A*,where A* denotes the conjugate transpose of A. Fromthe similarity theory of matrices, any idempotent matrix A   canbe decomposed as , where Ik isthe identity matrix of order k and k is the rank of A;any orthogonal projector A   can be decomposed as ,where U-1=U*. Idempotent matrices and orthogonalprojectors appear almost everywhere, and have been the objects of many studiesin matrix theory and its applications.