A Study on Rank Equalities For Idempotent Matrices

A square matrix A over the complex field C is said to be idempotent if A2=A; said to be an orthogonal projector if it is both idempotent and Hermitian, i.e., A2=A=A*, where A* denotes the conjugate transpose of A. From the similarity theory of matrices, any idempotent matrix A can be decomposed as , where Ik is the 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 orthogonal projectors appear almost everywhere, and have been the objects of many studies in matrix theory and its applications. Idempotent matrices and orthogonal projectors also have close links with generalized inverses of matrices. For instance, both AA- and A-A are idempotent for any generalized inverse A- of A ; both AA† and A†Aare orthogonal projectors for the Moore–Penrose inverse A† of A . Many other types of matrices can be converted into idempotent matrices through some elementary operations. For instance, if A2=-A, then (-A)2=-A, i.e., -A is idempotent; if A2=Im, then (Im±A)/2 are idempotent; if A2=-Im, then (Im±iA)/2are idempotent. In general, any matrix A satisfying a quadratic equation A2+aA+bIm=0 can be written as [A-(a/2)Im]2=(a2/4-b)Im. If a2/4-b≠0, then one can also write out an idempotent matrix from this equality. In the investigation of idempotent matrices and their applications, one often encounters various matrix expressions consisting of idempotent matrices. For example, PQ , P±Q, λ1P+λ2Q, PA-AQ, Im-PQ,PQ±QP, (PQ)2-PQ, AA†±A†A, AA-±B-B, where P and Q are two idempotent matrices. On the other hand, one also considers matrix decompositions associated with idempotent matrices, like A=P1±P2, A=P1P2±P2P1, A=P1⋯Pk; where P1,P2,…,Pk are idempotents. In such situations, it is of interest to give some basic properties of these matrix expressions, as well as relationships among these matrix expressions. When investigating these problems, we have noticed that the rank of matrix is a very rich technique for dealing with matrix expressions consisting of idempotent matrices. The rank of a matrix is invariant with respect to some basic operations for this matrix, such as, elementary matrix operations and similarity transformations. A well-known fact about matrix rank is: two matrices A and B of the same size are equivalent, i.e., there are two invertible matrices U and V such that UAV=B, if and only if r(A)=r(B). The simplest method for determining the linear independency of columns or rows of a matrix, as well as the rank of the matrix is to reduce the matrix to column or row echelon forms by elementary matrix operations. Theoretically, for any matrix expression consisting of idempotent matrices, one can establish some rank equalities associated with this expression. From these rank equalities, one can derive some basic properties on this expression. Rank formulas can be established through various block matrices and elementary block matrix operations. Some well-known results are given below:

Suppose P and Q are a pair of idempotents (including idempotent matrices over an arbitrary field F, idempotent operators on Banach and Hilbert spaces, idempotents in unital rings). The two fundamental operations for P and Q are given singularity, idempotency, tripotency and nilpotency of P±Q. Various results onP±Q and their properties can be found in the literature. Recall that a square matrix A of order m is nonsingular if and only if r(A)=m. If some rank equalities and inequalities for P±Q can be established, one can derive a variety of properties for P±Q from these rank equalities. A group of valuable rank equalities for a sum of two idempotent matrices are presented, but their proofs are omitted there. Here we restate these rank equalities and give their proofs.

Let P and Q be a pair of complex idempotent matrices of order m. Then the sum P+Qsatisfies the following rank equalities r(P+Q)=r[P-PQ,Q]=r[Q-QP,P], r(P+Q)=r(P-PQ-QP+QPQ)+r(Q), r(P+Q)=r(Q-PQ-QP+PQP)+r(P).

Recall that the rank of a matrix is an important invariant quantity under elementary matrix operations for this matrix, that is, these operations do not change the rank of the matrix. Thus, we first find by elementary block matrix operations the following trivial result: On the other hand, note P2=P and Q2=Q. By elementary block matrix operations, we also obtain another nontrivial rank equality Let P and Q be a pair of idempotent matrices of order m, and let a1and a2be two nonzero scalars such that a1+a2≠0.Then r(a1P+a2Q)=r(P+Q), that is, the rank of a1P+a2Qis invariant with respect to a1≠0,a2≠0and a1+a2≠0. Equality was proposed by Tian as a problem and solved by Bataille and other seven solvers. Equality can be derived from a result in Tian and Styan that for any P2=λP and Q2=λQ withλP≠0 and μ≠0,

Let P and Q be a pair of idempotent matrices of order m. Then equation

By elementary block matrix operations Also find by elementary block matrix operations

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