Miscellaneous equalities for idempotent matrices with applications

Abstract This article brings together miscellaneous formulas and facts on matrix expressions that are composed by idempotent matrices in one place with cogent introduction and references for further study. The author will present the basic mathematical ideas and methodologies of the matrix analytic theory in a readable, up-to-date, and comprehensive manner, including constructions of various algebraic matrix identities composed by the conventional operations of idempotent matrices, and uses of the block matrix method in the derivation of closed-form formulas for calculating the ranks of matrix expressions that are composed by idempotent matrices. The author also determines the maximum and minimum ranks of some matrix pencils composed by the products of matrices and their generalized inverses and uses the ranks to characterize algebraic performance of the matrix pencils.