# Upper and Lower Bounds for Ranks of the Matrix ExpressionX-XAX

Authors: Zhiping Xiong
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider the question of how to takeX such that the nonlinear matrix expression X-XAX attains
its maximal and minimal possible ranks.


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## Body

## 1. Introduction

Throughout this paperCm×n denotes the set of all m×n matrices over the complex field C. Ik denotes the identity matrix of order k and Om×n is the m×n matrix of all zero entries (if no confusion occurs, we will drop the subscript). For a matrix A∈Cm×n, A* and r(A) denote the conjugate transpose and the rank of the matrix A, respectively. (A,B) denotes a row block matrix consisting of A∈Cm×n and B∈Cm×k.LetA∈Cm×n; a generalized inverse X of A is a matrix which satisfies some of the following four Penrose equations [1]:
(1)AXA=A,XAX=X,(AX)*=AX,(XA)*=XA.For a subset{i,j,…,k} of the set {1,2,3,4}, the set of n×m matrices satisfying the equations (i),(j),…,(k) from (1) is denoted by A{i,j,…,k}. A matrix in A{i,j,…,k} is called an {i,j,…,k}-inverse of A and is denoted by A(i,j,…,k). For example, an n×m matrix X of the set A{2} is called a {2}-inverse of A and is denoted by X=A(2). The unique {1,2,3,4}-inverse of A is denoted by A†, which is called the Moore-Penrose inverse of A. For convenience, the symbols EA and FA stand for the two orthogonal projectors EA=Im-AA† and FA=In-A†A. We refer the reader to [2–4] for basic results on generalized inverses.In matrix theory and applications, there exists a nonlinear matrix expression that involves variable entries:(2)P(X)=X-XAX,
where A∈Cm×m is a given complex matrix and X∈Cm×m is a variable matrix. These nonlinear matrix expressions vary with respect to the choice of X. One of the fundamental problems for (2) is to determine the maximal and minimal possible ranks of the matrix expression P(X) when X is running over Cm×m. Since the rank of matrix is an integer between 0 and the minimum of row and column numbers of the matrix [5], then the maximal and minimal ranks of P(X) can be attained for some X.The investigation of extremal ranks of matrix expressions has many direct motivations in matrix analysis. For example, a matrix expressionD-AXB of order n is nonsingular if and only if the maximal rank of D-AXB with respect to X is n; two consistent matrix equations X1=X1AX1 and X2=X2BX2 have a common solution if and only if the minimal rank of the difference X1-X2 of their solutions is zero; a nonlinear matrix equation X=XAX is consistent if and only if the minimal rank of X-XAX with respect to X is zero. From the definition of the {2}-inverse of a matrix, we know that the solution X of the nonlinear matrix equation X=XAX is a {2}-inverse of matrix A; that is, using the minimal rank of X-XAX, we can find out the general expression of the {2}-inverses of a matrix A, which is a matrix such that the nonlinear matrix expression X-XAX attains its minimal rank. In general, for any two matrix expressions P(X1,X2,…,Xm) and Q(Y1,Y2,…,Yn) of the same size, there are X1,X2,…,Xm and Y1,Y2,…,Yn such that P(X1,X2,…,Xm)=Q(Y1,Y2,…,Yn) if and only if
(3)minX1,X2,…,Xm;Y1,Y2,…,Ynr(P(X1,X2,…,Xm)hhhhhhhhhhhhhhhh-Q(Y1,Y2,…,Yn))=0.These examples imply that the extremal ranks of matrix expressions have close links with many topics in matrix analysis and applications. Various statements on maximal and minimal ranks of matrix expressions are quite easy to understand for the people who know linear algebra. But the question now is how to give simple or closed forms for the extremal ranks of a matrix expression with respect to its variant matrices. The study on maximal and minimal ranks of matrix expression started in late 1980s. If want to know more about this question the reader can see [6–18].The work in this paper includes two parts. First, in Section2, we will consider how to choose a matrix X∈Cm×m, such that X-XAX has the maximal possible rank. Second, in Section 3, we will determine the minimal rank of X-XAX and present a general expression of the {2}-inverses of matrix A∈Cm×m.In order to find the extremal ranks of the nonlinear matrix expressionX-XAX, we need the following lemmas, which will be used in this paper.Lemma 1 (see [19]).
LetP(X1,X2)=A-B1X1C1-B2X2C2, where A∈Cm×n, B1∈Cm×p1, B2∈Cm×p2, C1∈Cq1×n, and C2∈Cq2×n are given matrices. Then for any variable matrices X1∈Cp1×q1 and X2∈Cp2×q2, one has
(4)maxX1,X2r(P(X1,X2))=min{r(A,B1,B2),r(AC1C2),hhhhhr(AB1C2O),r(AB2C1O)(AC1C2)},(5)minX1,X2r(P(X1,X2))=r(AC1C2)+r(A,B1,B2)+max{S1,S2},
where(6)S1=r(AB1C2O)-r(AB1B2C2OO)-r(AB1C1OC2O),S2=r(AB2C1O)-r(AB1B2C1OO)-r(AB2C1OC2O).Lemma 2 (see [20]).
LetA-BXC be a linear matrix expression over the complex field C, where A∈Cm×n, B∈Cm×k, and C∈Cl×n are given; X∈Ck×l is a variant matrix. Then the maximal rank of A-BXC with respect to X is
(7)maxXr(A-BXC)=min{r(A,B),r(AC)},
the general expression of X satisfying (7) is
(8)X=(EA2B)†EA2AFA1(CFA1)†+U,
where A1=EBA=(Im-BB†)A, A2=AFC=A(In-C†C)A, EA2=Im-A2A2†, and FA1=In-A1†A1, and the matrix U∈Ck×l is chosen such that
(9)r(EA2BUCFA1)=min{r(EA2B),r(CFA1)}.Lemma 3 (see [21]).
LetA∈Cm×n and B∈Cm×k. Then
(10)(A,B)†=((In+TT*)-1(A†-A†BC†)T*(In+TT*)-1(A†-A†BC†)+C†),
where C=(Im-AA†)B, T=A†B(Ik-C†C), and (In+TT*)-1=In-T(Ik+T*T)-1T*.Lemma 4 (see [22]).
LetA∈Cm×n, B∈Cm×k, and C∈Cl×n. Then
(11)r(A,B)=r(A)+r(EAB)=r(B)+r(EBA),r(AC)=r(A)+r(CFA)=r(C)+r(AFC),r(A+B)≤r(A)+r(B),
where EA=Im-AA†, EB=Im-BB†, FA=In-A†A and FC=In-C†C.


## 2. The Maximal Rank ofX-XAX with respect to X

LetA∈Cm×m be a given matrix; in this section, we will present the maximal rank of the nonlinear matrix expression X-XAX, with respect to the variable matrix X∈Cm×m. The relative results are included in the following three lemmas.Lemma 5.
LetA∈Cm×m, and I=Im denotes the identity matrix of order m. Then
(12)(IIOA)†=((I+FA)-1-A†FA(I+FA)-1A†).Proof.
By Lemma3 with M=(IO) and N=(IA), we have
(13)(IIOA)†=(M,N)†=((I+TT*)-1(M†-M†NK†)T*(I+TT*)-1(M†-M†NK†+K†)),
where
(14)M†=(I,O),K=(I-MM†)N=(OA),K†=(O,A†),(15)M†-M†NK†=(I,A†),T=M†N(I-K†K)=FA=I-A†A.
Combining (13), (14) with (15), we have
(16)(IIOA)†=((I+FA)-1-(I+FA)-1A†FA(I+FA)-1-FA(I+FA)-1A†+A†)=((I+FA)-1-A†FA(I+FA)-1A†).
The second equality holds as
(17)(I+FA)-1=I-FA(I+FA)-1FA,FAA†=(I-A†A)A†=O.Lemma 6.
LetA∈Cm×m, and I=Im denotes the identity matrix of order m. Then
(18)maxUr((OOOA)-(IIOA)U(O-A-IO))=m+r(A).
The general expression ofU satisfying (18) is
(19)U=(A†+V1V2-A†+V3V4),
where V1,V2,V3,V4∈Cm×m are chosen such that
(20)r((IIOA)(V1V2V3V4)(O-A-IO))=m+r(A).Proof.
By Lemma2 with P=(OOOA), Q=(IIOA), and W=(O-A-IO), we have
(21)maxUr((OOOA)-(IIOA)U(O-A-IO))=maxUr(P-QUW)=min{r(P,Q),r(PW)}=min{r(OOIIOAOA),r(OOOAO-A-IO)}=m+r(A).
From Lemmas2 and 5, we have the general expression of U satisfying (21) as
(22)U=(EA2Q)†EA2PFA1(WFA1)†+V,
where
(23)V=(V1V2V3V4),Q†=((I+FA)-1-A†FA(I+FA)-1A†),W†=(O-I-A†O),A1=EQP=(I-QQ†)P=(I-(IIOA)((I+FA)-1-A†FA(I+FA)-1A†))(OOOA)=(OOOEA)(OOOA)=O,(24)A2=PFW=P(I-W†W)=(OOOA)(I-(O-I-A†O)(O-A-IO))=(OOOA)(OOOFA)=O.
Combining (22), (23) with (24), we have EA2=I, FA1=I and
(25)U=Q†PW†+V=((I+FA)-1-A†FA(I+FA)-1A†)(OOOA)(O-I-A†O)+(V1V2V3V4)=(A†+V1V2-A†+V3V4),
where V1,V2,V3,V4∈Cm×m are chosen such that
(26)r(EA2Q(V1V2V3V4)WFA1)=r((IIOA)(V1V2V3V4)(O-A-IO))=min{r(EA2Q),r(WFA1)}=min{r(Q),r(W)}=m+r(A).Lemma 7.
LetA∈Cm×m. Then there exist some matrices V∈Cm×m, such that
(27)maxVr[(A†+V)-(A†+V)A(A†+V)]=m.Proof.
Applying Lemma4, we have
(28)r[(A†+V)-(A†+V)A(A†+V)]=r(A†-A†AA†+V-VA†A-A†AV-VAV)≤r(A†-A†AA†)+r(V-VA†A-A†AV-VAV)=0+r(V-VA†A-A†AV-VAV)=r(V-VAA†-A†AVVAAVA)-r(A)=r((OOOA)-(-IO)V(I-AA†,A)-(A†A-A)V(I,O))-r(A).
From (27) and (28), we have
(29)maxVr[(A†+V)-(A†+V)A(A†+V)]=maxVr((OOOA)-(-IO)V(I-AA†,A)-(A†A-A)V(I,O))-r(A).By Lemma 1, we have
(30)maxVr((OOOA)-(-IO)V(I-AA†,A)-(A†A-A)V(I,O))=min{r(OO-IA†AOAOA),r(OOOAI-AA†AIO),r(OO-IOAOIOO),r(OOA†AOA-AI-AA†AO)(OOOAI-AA†AIO)}=min{m+r(A),m+r(A),2m+r(A),m+r(A)}=m+r(A).
Combining (29) with (30), we get the result (27).According to Lemmas5, 6, and 7, we immediately obtain the following theorem.Theorem 8.
LetA∈Cm×m be a given matrix, and X∈Cm×m is a variant matrix. Then
(31)maxXr(X-XAX)=m.
In consequence,(1)
there always existsX∈Cm×m, such that X-XAX is nonsingular;(2)
the matricesX satisfying (31) are given by X=A†+V and V∈Cm×m is the same as in (27).Proof.
First describe a special congruence transformation for a block matrix, which reduces the calculation of the maximal rank ofX-XAX:
(32)maxXr(X-XAX)=maxX(XXAAXA)-r(A)=maxXr((OOOA)-(-I-A)X(I,O)-(IO)X(O-A))-r(A).
By Lemma1, we have
(33)maxXr((OOOA)-(-I-A)X(I,O)-(IO)X(O-A))=min{r(OO-IIOA-AO),r(OOOAIOO-A),r(OO-IOA-AO-AO),r(OOIOAOIOO)(OOOAIOO-A)}=min{m+r(A),m+r(A),m+r(A),2m+r(A)}=m+r(A).Combining (32) with (33), we have
(34)maxXr(X-XAX)=m.
On the other hand, from another special congruence transformation for a block matrix, we have(35)maxXr(X-XAX)=maxX(XXAAXA)-r(A)=maxXr((OOOA)-(IIOA)(XOOX)(O-A-IO))-r(A).
Combining formula (35) with Lemma 6, we have
(36)maxXr(X-XAX)=maxXr((OOOA)-(IIOA)(XOOX)(O-A-IO))-r(A)=min{r(OOIIOAOA),r(OOOAO-A-IO)}-r(A)=m+r(A)-r(A)=m.
That is, theres always exist X∈Cm×m, such that X-XAX is nonsingular.
From the results in Lemmas6 and 7, we obtain that there always exist X=A†+V, such that
(37)maxXr(X-XAX)=maxVr[(A†+V)-(A†+V)A(A†+V)]=m.


## 3. The Minimal Rank ofX-XAX with respect to X

In this section, we will present the minimal rank of the nonlinear matrix expressionX-XAX. Moreover, we will consider how to choose a matrix X, such that X-XAX has the minimal possible rank.Theorem 9.
LetA∈Cm×m be a given matrix, and X∈Cm×m is a variant matrix. Then
(38)minXr(X-XAX)=0.
In consequence, there exists X∈Cm×m, such that the nonlinear matrix equation X=XAX is consistent.Proof.
By formula (5) in Lemma 1, we have
(39)minXr(X-XAX)=minX(XXAAXA)-r(A)=maxXr((OOOA)-(-I-A)X(I,O)-(IO)X(O-A))-r(A)=r(OOOAO-A-IO)+r(OOIIOAOA)+max{S1,S2}-r(A)=2m+r(A)+max{S1,S2},
where
(40)S1=r(OOIOAO-IOO)-r(OOIIOAOA-IOOO)-r(OOIOAOO-AO-IOO)=-2m-r(A),(41)S2=r(OOIOAAO-AO)-r(OOIIOAOAO-AOO)-r(OOIOAAO-AO-IOO)=-2m-r(A).
Combining (39), (40) with (41), we have
(42)minXr(X-XAX)=0.Corollary 10.
LetA∈Cm×m be a given matrix. Then the matrix X∈Cm×m satisfying the matrix equation X=XAX is given by
(43)X=(A†+FAV)A(A†+WEA),
where V,W∈Cm×m are two variant matrices.Proof.
PuttingX=(A†+FAV)A(A†+WEA) into X-XAX yields
(44)X-XAX=(A†+FAV)A(A†+WEA)-(A†+FAV)AA†A(A†+FAV)A(A†+WEA)=(A†+FAV)A(A†+WEA)-(A†+FAV)AA†A(A†+WEA)=O.



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