# An Analog of the Adjugate Matrix for the Outer InverseAT,S(2)

Authors: Xiaoji Liu; Guangyan Zhu; Guangping Zhou; Yaoming Yu
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

We investigate the determinantal representation by exploiting the limiting expression for the generalized inverseAT,S(2). We show the equivalent relationship between the existence
and limiting expression of AT,S(2) and some limiting processes of matrices and deduce the new determinantal
representations of AT,S(2), based on some analog of the classical adjoint matrix. Using the
analog of the classical adjoint matrix, we present Cramer rules for the restricted matrix equation AXB=D,ℛ(X)⊂T,𝒩(X)⊃S~.


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## Body

## 1. Introduction

Throughout this paperℂm×n denotes the set of m×n matrices over the complex number field ℂ, and ℂrm×n denotes its subset in which every matrix has rank r. I stands for the identity matrix of appropriate order (dimension).LetA∈ℂm×n, and let M and N be Hermitian positive definite matrices of orders m and n, respectively. Consider the following equations:(1)AXA=A,(2)XAX=A,(3)(AX)*=AX,(3M)(MAX)*=MAX,(4)(XA)*=XA,(4N)(NXA)*=NXA.X is called a {2}- (or outer) inverse of A if it satisfies (2) and denoted by A(2). X is called the Moore-Penrose inverse of A if it satisfies (1), (2), (3), and (5) and denoted by A†. X is called the weighted Moore-Penrose inverse of A (with respect to M,N) if it satisfies (1), (2), (4), and (6) and denoted by AMN+ (see, e.g., [1, 2]).LetA∈ℂn×n. Then a matrix X satisfying(1k)AkXA=Ak,(2*)XAX=X,(5)AX=XA,
where k is some positive integer, is called the Drazin inverse of A and denoted by Ad. The smallest positive integer k such that X and A satisfy (7),  (8), and (9), then it is called the Drazin index and denoted by k=Ind(A). It is clear that Ind(A) is the smallest positive integer k satisfying rank(Ak)=rank(Ak+1) (see [3]). If k=1, then X is called the group inverse of A and denoted by Ag. As is well known, Ag exists if and only if rankA=rankA2. The generalized inverses, and in particular Moore-Penrose, group and Drazin inverses, have also been studied in the context of semigroups, rings of Banach and C* algebras (see [4–8]).In addition, if a matrixX satisfies (1) and (5), then it is called a {1,5}-inverse of A and is denoted by A(1,5).LetA∈ℂm×n, W∈ℂn×m. Then the matrix X∈ℂm×n satisfying(1kW)(AW)k+1XW=(AW)k,(2W)XWAWX=X,(5W)AWX=XWA,
where k is some nonnegative integer, is called the W-weighted Drazin inverse of A, and is denoted by X=Ad,W (see [9]). It is obvious that when m=n and W=In, X is called the Drazin inverse of A.Lemma 1.1 (see [1, Theorem  2.14]).
LetA∈ℂrm×n, and let T and S be subspaces of ℂn and ℂm, respectively, with dimT=dimS⊥=t≤r. Then A has a {2}-inverse X such that ℛ(X)=T and 𝒩(X)=S if and only if
(1.1)AT⊕S=Cm
in which case X is unique and denoted by AT,S(2).IfAT,S(2) exists and there exists a matrix G such that ℛ(G)=T and 𝒩(G)=S, then GAAT,S(2)=G and AT,S(2)AG=G.It is well known that several important generalized inverses, such as the Moore-Penrose inverseA†, the weighted Moore-Penrose inverse AM,N+, the Drazin inverse Ad, and the group inverse Ag, are outer inverses AT,S(2) for some specific choice of T and S, are all the generalized inverse AT,S(2), {2}- (or outer) inverse of A with the prescribed range T and null space S (see [2, 10] in the context of complex matrices and [11] in the context of semigroups).Determinantal representation of the generalized inverseAT,S(2) was studied in [12, 13]. We will investigate further such representation by exploiting the limiting expression for AT,S(2). The paper is organized as follows. In Section 2, we investigate the equivalent relationship between the existence of AT,S(2) and the limiting process of matrices limλ→0G(AG+λI)-1 or limλ→0(GA+λI)-1G and deduce the new determinantal representations of AT,S(2), based on some analog of the classical adjoint matrix, by exploiting limiting expression. In Section 3, using the analog of the classical adjoint matrix in Section 2, we present Cramer rules for the restricted matrix equation AXB=D,ℛ(X)⊂T,𝒩(X)⊃S̃. In Section 4, we give an example for solving the solution of the restricted matrix equation by using our expression. We introduce the following notations.For1≤k≤n, the symbol 𝒬k,n denotes the set {α:α=(α1,…,αk),1≤α1<⋯<αk≤n,whereαi,i=1,…,k,areintegers}. And 𝒬k,n{j}:={β:β∈𝒬k,n,j∈β}, where j∈{1,…,n}.LetA=(aij)∈ℂm×n. The symbols a.j and ai. stand for the jth column and the ith row of A, respectively. In the same way, denote by a.j* and ai.* the jth column and the ith row of Hermitian adjoint matrix A*. The symbol A.j(b) (or Aj.(b)) denotes the matrix obtained from A by replacing its jth column (or row) with some vector b (or bT). We write the range of A by ℛ(A)={Ax:x∈ℂn} and the null space of A by 𝒩(A)={x∈ℂn:Ax=0}. Let B∈ℂp×q. We define the range of a pair of A and B as ℛ(A,B)={AWB:W∈ℂn×p}.Letα∈𝒬k,m and β∈𝒬k,n, where 1≤k≤min{m,n}. Then |Aβα| denotes a minor of A determined by the row indexed by α and the columns indexed by β. When m=n, the cofactor of aij in A is denoted by ∂|A|/∂aij.


## 2. Analogs of the Adjugate Matrix forAT,S(2)

We start with the following theorem which reveals the intrinsic relation between the existence ofAT,S(2) and of limλ→0G(AG+λI)-1or limλ→0(GA+λI)-1G. Here λ→0 means λ→0 through any neighborhood of 0 in ℂ which excludes the nonzero eigenvalues of a square matrix. In [14], Wei pointed out that the existence of AT,S(2) implies the existence of limλ→0G(AG+λI)-1 or limλ→0(GA+λI)-1G. The following result will show that the converse is true under some condition.Theorem 2.1.
LetA∈ℂrm×n, and let T and S be subspaces of ℂn and ℂm, respectively, with dimT=dimS⊥=t≤r. Let G∈ℂrn×m with ℛ(G)=T and 𝒩(G)=S. Then the following statements are equivalent:(i)
AT,S(2) exists;(ii)
limλ→0G(AG+λI)-1 exists and rank(AG)=rank(G);(iii)
limλ→0(GA+λI)-1G exists and rank(GA)=rank(G).
In this case,(2.1)AT,S(2)=lim⁡λ→0⁡(GA+λI)-1G=lim⁡λ→0⁡G(AG+λI)-1.Proof.
(i)⇔(ii) Assume that AT,S(2) exists. By [14, Theorem  2.4], limλ→0G(AG+λI)-1 exists. Since G=AT,S(2)AG, rank(AG)=rank(G).
Conversely, assume thatlimλ→0G(AG+λI)-1 exists and rank(AG)=rank(G). So
(2.2)lim⁡λ→0⁡(AG+λI)-1AG=lim⁡λ→0⁡AG(AG+λI)-1
exists. By [15, Theorem], (AG)g exists. So (AG)(1,5) exists, and then, by [13, Theorem  2], AT,S(2) exists.
Similarly, we can show that (i)⇔(iii). Equation (2.1) comes from [14, equation (2.16)].Lemma 2.2.
LetA=(aij)∈ℂm×n and G=(gij)∈ℂtn×m. Then rank(GA).i(g.j)≤t, where 1≤i≤n, 1≤j≤m, and rank(AG)i.(gj.)≤t, where 1≤i≤m, 1≤j≤n.Proof.
LetPik(a) be an n×n matrix with a in the (i,k) entry, 1 in all diagonal entries, and 0 in others. It is an elementary matrix and
(2.3)(GA).i(g.j)∏k≠iPik(-ajk)=(∑k≠jg1kak1⋯g1j⋯∑k≠jg1kakn⋮⋮⋮⋮⋮∑k≠jgnkak1⋯gnj⋯∑k≠jgnkakn)ith=(g11⋯g1j⋯g1m⋮⋮⋮⋮⋮gi1⋯gij⋯gim⋮⋮⋮⋮⋮gn1⋯gnj⋯gnm)(a11⋯0⋯a1n⋮⋮⋮⋮⋮0⋯1⋯0⋮⋮⋮⋮⋮am1⋯0⋯amm)jth.ith
It follows from the invertibility of Pik(a),i≠k, that rank(GA).i(g.j)≤t.
Analogously, the inequationrank(AG)i.(gj.)≤t can be proved. So the proof is complete.Recall that iffA(λ)=det(λI+A)=λn+d1λn-1⋯+dn-1λ+dn is the characteristic polynomial of an n×n matrix—A over ℂ, then di is the sum of all i×i principal minors of A, where i=1,…,n (see, e.g., [16]).Theorem 2.3.
LetA,T,S, and G be the same as in Theorem 2.1. Write G=(gij). Suppose that the generalized inverse AT,S(2) of A exists. Then AT,S(2) can be represented as follows:
(2.4)AT,S(2)=(xijdt(GA))n×m,
where
(2.5)xij=∑β∈Qt,n{i}|((GA).i(g.j))ββ|,dt(GA)=∑β∈Qt,n|(GA)ββ|,
or
(2.6)AT,S(2)=(yijdt(AG))n×m,
where
(2.7)yij=∑α∈Qt,m{j}|((AG)j.(gi.))αα|,dt(AG)=∑α∈Qt,m|(AG)αα|.Proof.
We will only show the representation (2.5) since the proof of (2.7) is similar. If -λ is not the eigenvalue of GA, then the matrix λI+GA is invertible, and
(2.8)(λI+GA)-1=1det⁡⁡(λI+GA)(X11X21⋯Xn1X12X22⋯Xn2⋮⋮⋮⋮X1nX2n⋯Xnn),
where Xij,i,j=1,…,n, are cofactors of λI+GA. It is easy to see that
(2.9)∑l=1nXilglj=det⁡⁡(λI+GA).i(g.j).
So, by (2.1),
(2.10)AT,S(2)=lim⁡λ→0⁡(det⁡⁡(λI+GA).1(g.1)det⁡⁡(λI+GA)⋯det⁡⁡(λI+GA).1(g.m)det⁡⁡(λI+GA)⋮⋮⋮det⁡⁡(λI+GA).n(g.1)det⁡⁡(λI+GA)⋯det⁡⁡(λI+GA).n(g.m)det⁡⁡(λI+GA)).
We have the characteristic polynomial ofGA(2.11)fGA(λ)=det⁡⁡(λI+GA)=λn+d1λn-1+d2λn-2+⋯+dn,
where di(1≤i≤n) is a sum of i×i principal minors of GA. Since rank(GA)≤rank(G)=t, dn=dn-1=⋯=dt+1=0 and
(2.12)det⁡⁡(λI+GA)=λn+d1λn-1+d2λn-2+⋯+dtλn-t.
Expanding det(λI+GA).i(g.j), we have
(2.13)det⁡⁡(λI+GA).i(g.j)=x1(ij)λn-1+x2(ij)λn-2+⋯+xn(ij),
where xk(ij)=∑β∈𝒬k,n{i}|((GA).i(g.j))ββ|, 1≤k≤n, for 1≤i≤n and 1≤j≤m.
By Lemma2.2, rank(GA).i(g.j)≤t and so |((GA).i(g.j))ββ|=0, k>t and β∈𝒬k,n{i}, for all i,j. Therefore, xk(ij)=0, k≤n, for all i,j. Consequently,
(2.14)det⁡⁡(λI+GA).i(g.j)=x1(ij)λn-1+x2(ij)λn-2+⋯+xt(ij)λn-t.
Substituting (2.12) and (2.14) into (2.10) yields
(2.15)AT,S(2)=lim⁡λ→0⁡(x1(11)λn-1+⋯+xt(11)λn-tλn+d1λn-1+⋯+dtλn-t⋯x1(1m)λn-1+⋯+xt(1m)λn-tλn+d1λn-1+⋯+dtλn-t⋮⋮⋮x1(n1)λn-1+⋯+xt(n1)λn-tλn+d1λn-1+⋯+dtλn-t⋯x1(nm)λn-1+⋯+xt(nm)λn-tλn+d1λn-1+⋯+dtλn-t)=(xt(11)dt⋯xt(1m)dt⋮⋮⋮xt(n1)dt⋯xt(nm)dt).
Substitutingxij for xt(ij) in the above equation, we reach (2.5).Remark 2.4.
The proofs of Lemma2.2 and Theorem 2.3 are based on the general techniques and methods obtained previously by [17], respectively.Remark 2.5.
(i) By using (2.5), we can obtain (2.17) in [12, Theorem  2.3]. In fact, u=dt(GA) and, by the Binet-Cauchy formula,
(2.16)xij=∑β∈Qt,n{i}|((GA).i(g.j))ββ|=∑β∈Qt,n{i}∑kgkj∂|(GA)ββ|∂ski=∑β∈Qt,n,α∈Qt,m∑kgkj∂|Gβα|∂gkj∂|Aαβ|∂aji=∑α∈Qt,m,β∈Qt,ndet⁡⁡(Gβα)∂|Aαβ|∂aji,
where skj=(GA)kj. Note that ∂|Aαβ|/∂aij=0 if i∉α or j∉β. In addition, using the symbols in [13], we can rewrite (2.5) as [13, equation (13)] over ℂ.
(ii) This method is especially efficient whenGA or AG is given (comparing with [12, Theorem  2]).Observing the particular case from Theorem2.3, G=(gij)=N-1A*M, where M and N are Hermitian positive definite matrices, we obtain the following corollary in which the symbols g.j∶=(g).j and gi.∶=(g)i..Corollary 2.6.
LetA∈ℂrm×n and G=N-1A*M, where M and N are Hermitian positive definite matrices of order m and n, respectively, Then
(2.17)AMN†=(xijdr(GA))n×m,
where
(2.18)xij=∑β∈Qr,n{i}|((GA).i(g.j))ββ|,dr(GA)=∑β∈Qr,n|(GA)ββ|,
or
(2.19)AMN†=(yijdr(AG))n×m,
where
(2.20)yij=∑α∈Qr,m{j}|((AG)j.(gi.))αα|,dr(AG)=∑α∈Qr,m|(AG)αα|.IfM and N are identity matrices, then we can obtain the following result.Corollary 2.7 (see [17, Theorem  2.2]).
The Moore-Penrose inverseA† of A=(aij)∈ℂrm×n can be represented as follows:
(2.21)A†=(xijdr(A*A))n×m,
where
(2.22)xij=∑β∈Qr,n{i}|((A*A).i(a.j*))ββ|,dr(A*A)=∑β∈Qr,n|(A*A)ββ|,
or
(2.23)A†=(yijdr(AA*))n×m,
where
(2.24)yij=∑α∈Qr,m{j}|((AA*)j.(ai.*))αα|,dr(AA*)=∑α∈Qr,m|(AA*)αα|.Note thatAd,W=(WAW)ℛ((AW)kA),𝒩((AW)kA)(2). Therefore, when G=(AW)kA in Theorem 2.3, we have the following corollary.Corollary 2.8.
LetA∈ℂm×n, W∈ℂn×m, and k=max{Ind(AW),Ind(WA)}. If rank(AW)k=t, rank(WA)k=r, and (AW)kA=(cij)m×n, then
(2.25)Ad,W=(xijdt((AW)k+2))m×n,
where
(2.26)xij=∑β∈Qt,m{i}|(((AW)k+2).i(c.j))ββ|,dt((WA)k+2)=∑β∈Qt,m|((AW)k+2)ββ|,
or
(2.27)Ad,W=(yijdr((WA)k+2))m×n,
where
(2.28)yij=∑α∈Qr,n{j}|(((WA)k+2)j.(ci.))αα|,dr((WA)k+2)=∑α∈Qr,n|((WA)k+2)αα|.WhenG=Ak with k=Ind(A) in Theorem 2.3, we have the following corollary.Corollary 2.9 (see [17, Theorem  3.3]).
LetA∈ℂn×n with IndA=k and rankAk=r, and Ak=(aij(k))n×n. Then
(2.29)Ad=(xijdr(Ak+1))n×n,
where
(2.30)dij=∑β∈Qr,n{i}|((A.ik+1)(a.j(k)))ββ|,dr(Ak+1)=∑β∈Qr,n|(Ak+1)ββ|.Finally, we turn our attention to the two projectorsAT,S(2)A and AAT,S(2). The limiting expressions for AT,S(2) in (2.1) bring us the following:(2.31)AT,S(2)A=lim⁡λ→0⁡(GA+λI)-1GA,AAT,S(2)=lim⁡λ→0⁡AG(AG+λI)-1.Corollary 2.10.
LetA,T,S, and G be the same as in Theorem 2.1. Write GA=(sij) and AG=(hij). Suppose that AT,S(2) exists. Then AT,S(2)A of AAT,S(2) can be represented as follows:
(2.32)AT,S(2)A=(xijdt(GA))n×n,
where
(2.33)xij=∑β∈Qt,n{i}|((GA).i(s.j))ββ|,dt(GA)=∑β∈Qt,n|(GA)ββ|,AAT,S(2)=(yijdt(AG))m×m,
where
(2.34)yij=∑α∈Qt,m{j}|((AG)j.(hi.))αα|,dt(AG)=∑α∈Qt,m|(AG)αα|.


## 3. Cramer Rules for the Solution of the Restricted Matrix Equation

The restricted matrix equation problem is mainly to find solution of a matrix equation or a system of matrix equations in a set of matrices which satisfy some constraint conditions. Such problems play an important role in applications in structural design, system identification, principal component analysis, exploration, remote sensing, biology, electricity, molecular spectroscopy, automatics control theory, vibration theory, finite elements, circuit theory, linear optimal, and so on. For example, the finite-element static model correction problem can be transformed to solve some constraint condition solution and its best approximation of the matrix equationAX=B. The undamped finite-element dynamic model correction problem can be attributed to solve some constraint condition solution and its best approximation of the matrix equation ATXA=B. These motivate the gradual development of theory in respect of the solution to the restricted matrix equation in recent years (see [18–27]).In this section, we consider the restricted matrix equation(3.1)AXB=D,R(X)⊂T,N(X)⊃S̃,
where A∈ℂrm×n, B∈ℂr̃p×q, D∈ℂm×q, T⊂ℂn, S⊂ℂm, T̃⊂ℂq and S̃⊂ℂp, satisfy(3.2)dim⁡⁡(T)=dim⁡⁡(S⊥)=t≤r,dim⁡⁡(T̃)=dim⁡⁡(S̃⊥)=t̃≤r̃.
Assume that there exist matrices G∈ℂn×m and G̃∈ℂq×p satisfying(3.3)R(G)=T,N(G)=S,R(G̃)=T̃,N(G̃)=S̃.
If AT,S(2) and BT̃,S̃(2) exist and D∈ℛ(AG,G̃B), then the restricted matrix equation (3.1) has the unique solution(3.4)X=AT,S(2)DBT̃,S̃(2)
(see [2,  Theorem  3.3.3] for the proof).In particular, whenD is a vector b and B=G̃=I1, the restricted matrix equation (3.1) becomes the restricted linear equation(3.5)Ax=b,x∈T.
If b∈Aℛ(G), then x=AT,S(2)b is the unique solution of the restricted linear equation (3.5) (see also [10,  Theorem  2.1]).Theorem 3.1.
GivenA,B,D=(dij),G=(gij),G̃=(g̃ij),T,S,T̃, and S̃ as above. If AT,S(2) and BT̃,S̃(2) exist and D∈ℛ(AG,G̃B), then X=AT,S(2)DBT̃,S̃(2) is the unique solution of the restricted matrix equation (3.1) and it can be represented as
(3.6)xij=∑k=1m∑β∈Qt,n{i},α∈Qt̃,p{j}|((GA).i(g.k))ββ||((BG̃)j.(f̃k.))αα|dt(GA)dt̃(BG̃)
or
(3.7)xij=∑k=1q∑β∈Qt,n{i},α∈Qt̃,p{j}|((GA).i(f.k))ββ||((BG̃)j.(g̃k.))αα|dt(GA)dt̃(BG̃),
where f̃k.=dk.G̃ and f.k=Gd.k, i=1,…,n, and j=1,…,p.Proof.
By the argument above, we haveX=AT,S(2)DBT̃,S̃(2) is the unique solution of the restricted matrix equation (3.1). Setting Y=DBT̃,S̃(2) and using (2.7), we get that
(3.8)ykj=∑h=1qdkh(BT̃,S̃(2))hj=∑h=1qdkh∑α∈Qt̃,p{j}|((BG̃)j.(g̃h.))αα|dt̃(BG̃)=∑α∈Qt̃,p{j}|((BG̃)j.(∑h=1qdkhg̃h.))αα|dt̃(BG̃)=∑α∈Qt̃,p{j}|((BG̃)j.(f̃k.))αα|dt̃(BG̃),
where f̃k.=dk.G̃. Since X=AT,S(2)Y, by (2.5),
(3.9)xij=∑k=1m(AT,S(2))ikykj=∑k=1m∑β∈Qt,n{i}|((GA).i(g.k))ββ|dt(GA)∑α∈Qt̃,p{j}|((BG̃)j.(f̃k.))αα|dt̃(BG̃).
Hence, we have (3.6).
We can obtain (3.7) in the same way.In particular, whenD is a vector b and B=G̃=I1 in the above theorem, we have the following result from (3.7).Theorem 3.2.
GivenA,G,T, and S as above. If b∈Aℛ(G), then x=AT,S(2)b is the unique solution of the restricted linear equation Ax=b, x∈T, and it can be represented as
(3.10)xi=∑β∈Qt,n{i}|((GA).i(f))ββ|dt(GA),j=1,…,n,
where f=Gb.Remark 3.3.
Using the symbols in [13], we can rewrite (3.10) as [13, equation (27)].


## 4. Example

Let(4.1)A=(12220211004200210000),B=(211001),D=(01-10000000),G=(1-1200021000000000000),G̃=(100020).
Obviously, rankA=3, dimAT=dimT=2, and(4.2)T=R(G)⊂C4,S=N(G)⊂C5,T̃=R(G̃)⊂C2,S̃=N(G̃)⊂C3.
It is easy to verify that AT⊕S=ℂ5 and BT̃⊕S̃=ℂ3. Thus, AT,S(2) and BT̃,S̃(2) exist by Lemma 1.1.Now consider the restricted matrix equation(4.3)AXB=D,R(X)⊂T,N(X)⊃S̃.
Clearly,(4.4)AG=(1340004200000000000000000),G̃B=(2120),
and it is easy to verify that ℛ(D)⊂ℛ(AG) and 𝒩(D)⊃𝒩(G̃B) hold.Note thatℛ(D)⊂ℛ(AG) and 𝒩(D)⊃𝒩(G̃B) if and only if D∈ℛ(AG,G̃B). So, by Theorem 3.1, the unique solution of (4.3) exists.Computing(4.5)GA=(1095046400000000),BG̃=(220100020),d2(GA)=|1004|+|1900|+|1500|+|4600|+|4400|+|0000|=4,d2(BG̃)=|2210|+|2000|+|0020|=-2,f̃=(1-1200021000000000000)(01-10000000)=(11-200000),
and setting yik=∑β∈𝒬t,n{i}|((GA).i(f̃.k))ββ|, we have Table 1.Table 1yiki=1i=2i=3i=4k=14−200k=24000Similarly, settingzkj=∑α∈𝒬t̃,p{j}|((BG̃)j.(g̃k.))αα|, we have Table 2.Table 2zkjj=1j=2j=3k=10−20k=2−240So, by (3.7), we have(4.6)X=(xij)=(1-100-120000000).



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