In this paper, we discuss different characteristics of the BT-inverse of a square matrix introduced by Baksalary and Trenkler [On a generalized core inverse, Appl. Math. Comput., 236 (2014), 450-457]. While the BT-inverse is defined by a expression, we present some necessary and sufficient conditions for a matrix to be the BT-inverse. Then we give a canonical form of BT-inverse and investigate the relationships between BT-inverse and other generalized inverses by Core-EP decomposition. Some properties of BT-inverse concerned with some classes of special matrix are identified by Core-EP decomposition. Furthermore new representations of BT-inverse are given by the maximal classes of matrices.
Citation: Wanlin Jiang, Kezheng Zuo. Revisiting of the BT-inverse of matrices[J]. AIMS Mathematics, 2021, 6(3): 2607-2622. doi: 10.3934/math.2021158
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In this paper, we discuss different characteristics of the BT-inverse of a square matrix introduced by Baksalary and Trenkler [On a generalized core inverse, Appl. Math. Comput., 236 (2014), 450-457]. While the BT-inverse is defined by a expression, we present some necessary and sufficient conditions for a matrix to be the BT-inverse. Then we give a canonical form of BT-inverse and investigate the relationships between BT-inverse and other generalized inverses by Core-EP decomposition. Some properties of BT-inverse concerned with some classes of special matrix are identified by Core-EP decomposition. Furthermore new representations of BT-inverse are given by the maximal classes of matrices.
For many different generalized inverses such as A†,AD, ,
, AD,†,A(B,C),
below can all be characterized by several equations respectively, while there is no such equations to define A♢. Our main aim is to develop some necessary and sufficient conditions for a matrix to be the BT-inverse by equations and derive some properties of the BT-inverse.
Throughout this paper, we denote the set of m×n complex matrices by Cm×n. We denote the identity matrix of order n by In, the range space, the null space, the conjugate transpose and the rank of the matrix A∈Cm×n by R(A), N(A), A∗ and r(A), respectively. The index of A∈Cn×n, denoted by Ind(A), is the smallest nonnegative integer k such that r(Ak)=r(Ak+1). PL,M stands for the projector (idempotent) on the space L along the M. For A∈Cm×n, PA represents the orthogonal projection onto R(A), i.e. PA=PR(A)=AA†.
For the readers' convenience, we will first recall the definitions of some generalized inverses. For A∈Cm×n, the Moore-Penrose inverse A† of A is the unique matrix X∈Cn×m satisfying the following four Penrose equations [1]:
(1) AXA=A, (2) XAX=X, (3) (AX)∗=AX, (4) (XA)∗=XA. |
A matrix X∈Cn×m that satisfies condition (1) above is called an inner inverse of A and is denoted by A(1). A matrix X∈Cn×m that satisfies condition (2) above is called an outer inverse of A and is denoted by A(2). A matrix X∈Cn×m that satisfies condition (1) and condition (3) above is denoted by A(1,3). The symbol A{1}, A{1,3} stand for the set of all A(1), A(1,3) respectively. Let A∈Cm×n be of rank r, and T, S be a subspace of Cn,Cm where T, S is of dimension t (⩽r), m−t, respectively. Then a matrix X satisfies X=XAX, R(X)=T and N(X)=S if and only if AT⊕S=Cm, and in this case X denoted by A(2)T,S is unique.
The Drazin inverse of A∈Cn×n with Ind(A)=k, denoted by AD [2], is the unique matrix X∈Cn×n satisfying:
XAX=X, AX=XA, XAk+1=Ak. |
Especially, if Ind(A)=1, then the Drazin inverse of A is called the group inverse of A and is denoted by A#.
Baksalary and Trenkler [3] introduced the core inverse on the CCMn (CCMn={A|A∈Cn×n,r(A)=r(A2)}): the core inverse of A∈CCMn is defined to be the unique matrix X∈Cn×n such that
AX=PA, R(X)⊆R(A) |
and denoted by (see [3,4,5,6]).
Moreover, three kinds of generalizations of the core inverse were given for n×n complex matrices, called core-EP inverse, DMP-inverse and BT-inverse, respectively.
Firstly, for A∈Cn×n with Ind(A)=k, the unique matrix X∈Cn×n satisfying:
XAX=X, R(X)=R(X∗)=R(Ak), |
is called the Core-EP inverse of A written as (see [7,8,9,10]). Moreover, it is seen that
=(Ak+1(Ak)†)† (see [7,Theorem 2.7]).
Secondly, the DMP-inverse of A∈Cn×n with Ind(A)=k, written by AD,† [11,12], is defined as the unique matrix A∈Cn×n satisfying:
XAX=X, XA=ADA, AkX=AkA†. |
Moreover, it was proved that AD,†=ADAA†. Also, the dual DMP inverse of A was introduced in [12], namely A†,D=A†AAD.
Thirdly, the BT-inverse of A∈Cn×n, denoted by A♢[13], is defined as
A♢=(A2A†)†=(APA)†. |
In recent years, some new generalized inverses are introduced. The (B, C)-inverse of A∈Cm×n, denoted by A(B,C) [14,15], is the unique matrix X∈Cn×m satisfying:
XAB=B, CAX=C, R(X)=R(B), N(X)=N(C), |
where B,C∈Cn×m.
In [16], Wang and Chen introduced a new generalized inverse called the weak group inverse of A∈Cn×n, denoted by . It is defined as the unique matrix X∈Cn×n satisfying:
![]() |
Moreover, it is proved that = (
)2A.
While the authors in [13] introduced the BT-inverse defined as A♢=(APA)†, the characterizations of how a matrix is A♢, however, seldom gave. In this paper, we concern more on the necessary and sufficient conditions for a matrix to be A♢ and characterize the relationships between A♢ and other generalized inverses. The research is as follows. In Section 2, some indispensable matrix classes and lemmas are given. In Section 3, some characterizations of A♢ are given too. In Section 4, we first derive a canonical form of A♢ by Core-EP decomposition and verify the validity of it by Example 1. By the canonical form of A♢ and Core-EP decomposition, we obtain the relationships between A♢ and other generalized inverses and some properties of A♢ when A♢ or A belongs to some special matrix classes. In Section 5, we extend the representation A♢=(APA)† to a more general one by the maximal classes of matrices.
For convenience, some matrix classes will be given as follows.
These symbols CCMn, CPn, COPn and CEPn will stand for the subsets of Cn×n consisting of core matrices, projectors (idempotent matrices), orthogonal projectors (Hermitian idempotent matrices) and EP (Range-Hermitian) matrices, respectively, i.e.,
CCMn={A|A∈Cn×n,r(A2)=r(A)},CPn={A|A∈Cn×n,A2=A},COPn={A|A∈Cn×n,A2=A=A∗}={A|A∈Cn×n,A2=A=A†},CEPn={A|A∈Cn×n,AA†=A†A}={A|A∈Cn×n,R(A)=R(A∗)}. |
In order to present some characterizations and properties of A♢, we need to introduce the following lemmas.
Lemma 2.1. [17] Let A∈Cn×n, r(A)=r. Then we have
A=U[ΣKΣL00]U∗, | (2.1) |
where U∈Cn×n is unitary, Σ=diag(σ1,σ2,…,σr) is the diagonal matrix of singular values of A, σi>0(i=1,2,⋯,r) and K∈Cr×r, L∈Cr×(n−r) satisfy
KK∗+LL∗=Ir. | (2.2) |
Moreover, from (2.1), it follows that
A†=U[K∗Σ−10L∗Σ−10]U∗, PA=AA†=U[Ir000]U∗. | (2.3) |
AD=U[(ΣK)D((ΣK)D)2ΣL00]U∗, | (2.4) |
A♢=U[(ΣK)†000]U∗ | (2.5) |
and
![]() |
(2.6) |
The lemma below gives the Core-EP decomposition introduced by Wang which plays an important role in this paper.
Lemma 2.2. [9] Let A∈Cn×n with Ind(A)=k. Then there exists a unitary matrix U∈Cn×n such that
A=A1+A2=U[TS0N]U∗, | (2.7) |
A1=U[TS00]U∗, A2=U[000N]U∗, |
where T∈Ct×t is nonsingular with t=r(T)=r(Ak) and N is nilpotent of index k.
Lemma 2.3. [18,Lemma 6] Let A∈Cn×n with Ind(A)=k be the form of (2.7). Then
A†=U[T∗△−T∗△SN†(In−t−N†N)S∗△N†−(In−t−N†N)S∗△SN†]U∗, | (2.8) |
where N is not necessary nilpotent, △=(TT∗+S(In−t−N†N)S∗)−1, t=r(Ak).
From (2.7) and (2.8), a straightforward computation shows that
AA†=U[It00NN†]U∗, | (2.9) |
A†A=U[T∗△TT∗△S(In−t−N†N)(In−t−N†N)S∗△TN†N+(In−t−N†N)S∗△S(In−t−N†N)]U∗. | (2.10) |
Lemma 2.4. [13,Theorem 1] Let A∈Cn×n. Then
AA♢=PAPA, A♢A=PR(PAA∗),N((APA)†A). | (2.11) |
It is well-known that some of generalized inverses such as MP-inverse, Drazin inverse, DMP-inverse, etc. can be presented as an outer inverse under the condition of prescribed range and null space. Therefore, we will prove that the same holds in the case of BT-inverse as follows. In the following theorem, we show the other characterizations of BT-inverse by the fact that A♢AA♢=A♢.
Theorem 3.1. Let A,X∈Cn×n. Then the following conditions are equivalent:
(a) X=A♢;
(b) XAX=X, R(X)=R(PAA∗) and N(X)=N(PAA∗), i.e., X=A(2)R(PAA∗),N(PAA∗);
(c) XAX=X, AX=A(APA)† and XA=(APA)†A;
(d) XAX=X, AX=PAPA and XA=(APA)†A.
Proof. (a)⇒(b). From the definition of BT-inverse and Lemma 2.4, we derive that
A(APA)†=AA♢=PAPA, | (3.1) |
moreover
(APA)†A(APA)†=(APA)†APA(APA)†. | (3.2) |
From the definition of BT-inverse and (3.2), it follows that
A♢AA♢=(APA)†A(APA)†=(APA)†APA(APA)†=(APA)†=A♢, |
R(A♢)=R((APA)†)=R((APA)∗)=R(PAA∗), |
N(A♢)=N((APA)†)=N((APA)∗)=N(PAA∗). |
(b)⇒(c). From [19,Remark 3.1], we have that A(2)R(A♢),N(A♢) exits. It is easy to check that A♢=A(2)R((APA)†),N((APA)†)=A(2)R(PAA∗),N(PAA∗). Since X=A(2)R(PAA∗),N(PAA∗) and the uniqueness of X, we obtain that X=A♢. Then the rest of proof is trivial.
(c)⇒(d). Since AX=A(APA)†, by (3.1), we obtain that AX=APA(APA)†=PAPA.
(d)⇒(a). By the condition, we conclude that
X=XAX=XAPA(APA)†=(APA)†APA(APA)†=(APA)†=A♢. |
In the following theorem, we present a connection between (B, C)-inverse and BT-inverse showing that a BT-inverse of a matrix A∈Cn×n is its (PAA∗,PAA∗)-inverse.
Theorem 3.2. Let A∈Cn×n. Then A♢=A(PAA∗,PAA∗).
Proof. From the definition of BT-inverse and (3.1), it follows that
A♢APAA∗=(APA)†APA(APA)∗=(APA)∗, |
PAA∗AA♢=(APA)∗A(APA)†=(APA)∗(APA)(APA)†=(APA)∗, |
R(A♢)=R(PAA∗),N(A♢)=N(PAA∗). |
Hence A♢=A(PAA∗,PAA∗).
According to the fact that R(A♢)=R(PAA∗) and N(A♢)=N(PAA∗), there are several different characterizations of BT-inverse as follows.
Theorem 3.3. Let A,X∈Cn×n. Then the following conditions are equivalent:
(a) X=A♢;
(b) AX=A(APA)†, R(X)=R(PAA∗);
(c) AX=PAPA, R(X)=R(PAA∗);
(d) PAX=(APA)†, R(X)=R(PAA∗);
(e) A†X=A†(APA)†, R(X)=R(PAA∗);
(f) XA=(APA)†A, N(X)=N(PAA∗);
(g) XA=PR(PAA∗),N((APA)†A), N(X)=N(PAA∗).
Proof. That (a) implies all other items (b), (c), (d), (e), (f) and (g) can be checked directly by Theorem 3.1, the definition of BT-inverse and Lemma 2.4.
(b)⇒(a). By R(X)=R(PAA∗), we have X=(APA)†T for some T∈Cn×n. By (3.2), then
X=(APA)†T=(APA)†APA(APA)†T=(APA)†AX=(APA)†A(APA)†=(APA)†APA(APA)†=A♢. |
(c)⇒(b). Since AX=PAPA, by (3.1), we obtain that AX=PAPA=APA(APA)†=A(APA)†.
(d)⇒(a). By R(X)=R(PAA∗), we get X=(APA)†T for some T∈Cn×n. By (3.2), then
X=(APA)†T=(APA)†APA(APA)†T=(APA)†APAX=(APA)†A(APA)†=(APA)†APA(APA)†=A♢. |
(e)⇒(d). Premultiplying A†X=A†(APA)† by A, we obtain that PAX=PA(APA)†=(APA)†.
(f)⇒(a). By N(X)=N(PAA∗), we obtain X=K(APA)† for some K∈Cn×n. By (3.2), then
X=K(APA)†=K(APA)†A(APA)†=XA(APA)†=(APA)†A(APA)†=(APA)†=A♢. |
(g)⇒(a). Since XA=PR(PAA∗),N((APA)†A)=PR((APA)†),N((APA)†A), we get XA(APA)†=(APA)†. By N(X)=N(PAA∗), we have X=K(APA)† for some K∈Cn×n. Then
X=K(APA)†=K(APA)†A(APA)†=XA(APA)†=A♢. |
Remark 3.4. Notice that the condition R(X)=R(PAA∗) in items (b), (c), (d) and (e) of Theorem 3.3 can be replaced by R(X)⊆R(PAA∗). Also the condition N(X)=N(PAA∗) in items (f),(g) of Theorem 3.3 can be replaced by N(PAA∗)⊆N(X).
Theorem 3.5. Let A,X∈Cn×n. Then the following conditions are equivalent:
(a) X=A♢;
(b) r(X)=r(A2), XA(APA)∗=(APA)∗ and AX=A(APA)†;
(c) r(X)=r(A2), (APA)∗AX=(APA)∗ and XA=A(APA)†A;
(d) r(X)=r(A2), XA(APA)∗=(APA)∗ and AX=PAPA;
(e) r(X)=r(A2), XA(APA)∗=(APA)∗ and PAX=(APA)†;
(f) r(X)=r(A2), XA(APA)∗=(APA)∗ and A†X=A†(APA)†;
(g) r(X)=r(A2), (APA)∗AX=(APA)∗ and XA=PR(PAA∗),N(PAA∗).
Proof. (a)⇒(b). For X=A♢, we get that r(A♢)=r(APA). For R(A2)=R(APAA)⊆R(APA)⊆R(A2), then we get that R(APA)=R(A2), hence r(A♢)=r(APA)=r(A2). From the definition of BT-inverse and the latter half of (2.11), we derive that A♢A(APA)∗=(APA)∗ and AA♢=A(APA)†.
That (a) implies all other items (c), (d), (e), (f) and (g) can be similarly proved.
(b)⇒(a). Combining r(X)=r(A2)=r(APA) with XA(APA)∗=(APA)∗, we obtain R(X)=R(PAA∗). Hence it follows from (b) of Theorem 3.3 that X=A♢.
(c)⇒(a). From r(X)=r(A2)=r(APA) and (APA)∗AX=(APA)∗, we get N(X)=N(PAA∗). Hence we get X=A♢ by (f) of Theorem 3.3.
The proofs of (d)⇒(a), (e)⇒(a) and (f)⇒(a) are analogous to that of (b)⇒(a). Also (g)⇒(a) follows similarly as in the part (c)⇒(a).
In this section, we first give the canonical form of BT-inverse by using Core-EP decomposition. Then some properties of BT-inverse will be given by utilizing the definition and the canonical form of BT-inverse.
Theorem 4.1. Let A∈Cn×n be of the form (2.7). Then
A♢=U[T∗△−T∗△SN♢(PN−PN♢)S∗△N♢−(PN−PN♢)S∗△SN♢]U∗, | (4.1) |
where △=[TT∗+S(PN−PN♢)S∗]−1.
Proof. By (2.9) of Lemma 2.3, we get that
A♢=(APA)†=(U[TSPN0NPN]U∗)†=U[TSPN0NPN]†U∗. |
From (2.8) of Lemma 2.3, we have that
A♢=U[T∗△−T∗△SPNN♢(PN−PN♢)S∗△N♢−(PN−PN♢)S∗△SPNN♢]U∗, |
where △=[TT∗+S(PN−PNPN♢)S∗]−1.
It is easy to check that PNN♢=N♢ by (2.3) and (2.5). Hence
A♢=U[T∗△−T∗△SN♢(PN−PN♢)S∗△N♢−(PN−PN♢)S∗△SN♢]U∗, |
where △=[TT∗+S(PN−PNPN♢)S∗]−1=[TT∗+S(PN−PN♢)S∗]−1.
Next, we will verify the correctness of the expression (4.1) as follows.
Example 1. Given matrix
A=[0.51910.59220.80960.33410.74910.08010.36640.69880.18340.19870.38970.28280.50730.65341.15330.10980.58470.73250.9618−0.17291.16830.39830.51910.34540.50720.3863−0.03721.05680.55830.33110.81770.31131.01330.74510.67380.57830.07140.15840.05240.11950.82940.33710.82220.98301.4529−0.1282−0.02990.35070.70320.51010.71890.02000.80320.58230.59890.57930.42540.09080.49430.90900.59230.61930.56850.49650.40730.31210.16420.24140.39790.33851.1399−0.04330.06940.60840.71490.80390.24170.34850.46290.34360.38830.36240.95900.48110.58950.29800.35990.40590.34570.49830.40630.37630.22830.74861.00070.81140.47960.3602−0.10580.5583]. |
By the definition of BT-inverse, it turns out that
r1=(APA)†=[1.2507−0.0226−0.0663−0.60580.21540.27900.04481.1114−0.8224−1.15970.20730.1052−0.0244−0.69520.0287−0.43451.7112−0.1370−0.2799−0.03480.01400.05400.06360.7072−0.14730.2573−0.5076−0.43220.5430−0.3668−1.36340.07400.12600.70410.1529−0.42190.4008−0.50870.31990.69530.56190.1288−0.2313−0.27230.5275−0.0635−0.66960.2278−0.28540.1209−1.15760.16230.32740.7051−0.6261−0.15630.1071−0.22720.45540.70881.53610.6173−0.7601−0.9706−0.44800.65440.00550.8060−0.3946−0.7210−0.2661−0.26180.76750.4571−0.2999−0.3593−0.7283−0.55810.51520.8686−0.76390.1845−0.20220.2158−0.19600.12040.9133−0.06000.1465−0.0297−0.0314−0.74610.1217−0.59910.27600.52320.0696−0.24710.25610.4946]. |
Assume that A is of the form (2.7), we obtain that
U=[0.29220.35670.25930.34270.02530.22890.6603−0.0103−0.3353−0.03230.3330−0.48010.13810.42010.20870.3648−0.05410.14850.4849−0.16220.33160.22410.3288−0.41950.1860−0.22290.0996−0.47650.4440−0.19340.2955−0.1610−0.21120.08920.25130.1539−0.3538−0.5646−0.5254−0.16550.38240.1840−0.2339−0.1527−0.72450.4114−0.1367−0.06630.14550.05900.33270.1261−0.60050.13600.0405−0.46350.12750.25840.0703−0.43580.26490.24880.0699−0.45040.40740.3044−0.24700.5568−0.1739−0.01870.2975−0.5624−0.1892−0.39280.0290−0.09600.44270.0202−0.12030.42930.30120.2960−0.05140.34640.1693−0.2581−0.2645−0.02150.13890.71700.3164−0.22490.54990.0559−0.3689−0.4342−0.25320.2237−0.2986−0.1258],T=[4.96950.59550.0256−0.1136−0.50710.49290.5074−1.05390−0.37450.7615−0.11750.0914−0.14660.0771−0.23350−0.6028−0.3745−0.16230.05360.36000.23170.3098000−0.68360.10550.1977−0.5123−0.050100000.6185−0.26330.40030.195300000.03920.6185−0.0897−0.55580000000.37050.2909000000−0.62300.3705], |
S=[−0.39730.09620.4349−0.04310.17270.13830.40680.03750.24370.01320.42050.5983−0.14540.3339−0.0429−0.0343],N=[0100]. |
According to (4.1), a straightforward computation shows that
A♢=[1.2507−0.0226−0.0663−0.60580.21540.27900.04481.1114−0.8224−1.15970.20730.1052−0.0244−0.69520.0287−0.43451.7112−0.1370−0.2799−0.03480.01400.05400.06360.7072−0.14730.2573−0.5076−0.43220.5430−0.3668−1.36340.07400.12600.70410.1529−0.42190.4008−0.50870.31990.69530.56190.1288−0.2313−0.27230.5275−0.0635−0.66960.2278−0.28540.1209−1.15760.16230.32740.7051−0.6261−0.15630.1071−0.22720.45540.70881.53610.6173−0.7601−0.9706−0.44800.65440.00550.8060−0.3946−0.7210−0.2661−0.26180.76750.4571−0.2999−0.3593−0.7283−0.55810.51520.8686−0.76390.1845−0.20220.2158−0.19600.12040.9133−0.06000.1465−0.0297−0.0314−0.74610.1217−0.59910.27600.52320.0696−0.24710.25610.4946]. |
Let ∥⋅∥ be the Frobenius norm, then it follows that
∥A♢−r1∥=3.5313×10−14 |
which implies the validity of the representation (4.1).
Lemma 4.2. [20] Let A∈Cn×n written as in (2.7). Then
AD=U[T−1(Tk+1)−1˜T00]U∗, | (4.2) |
where ˜T=k−1∑j=0TjSNk−1−j.
In [13], the necessary and sufficient conditions for A♢=A†, were given by using the Hartwig-Spindelböck decomposition in Lemma 2.1. We will prove the conditions that A♢=AD, A♢=A†,D and A♢=
are equivalent by utilizing Core-EP decomposition as follows.
Theorem 4.3. Let A∈Cn×n be decomposed by (2.7). Then the following statements are equivalent:
(a) S=0 and N2=0;
(b) A♢=AD;
(c) A2∈CEPn;
(d) A♢=A†,D;
(e) A♢=.
Proof. (a)⟺(b). It follows from the definition of A♢, Lemma 2.3 and (4.2).
A♢=AD⟺A2A†=(AD)†⟺U[TSPN0NPN]U∗=(U[T−1(Tk+1)−1˜T00]U∗)†⟺˜T=0, SPN=0, NPN=0⟺S=0, N2=0. |
(a)⟺(c). From (2.7) and (2.8), we can calculate that
A2=U[T2TS+SN0N2]U∗, |
(A2)†=U[(T2)∗△′−(T2)∗△′(TS+SN)(N2)†(In−t−(N2)†N2)(TS+SN)∗△′(N2)†−(In−t−(N2)†N2)(TS+SN)∗△′(TS+SN)(N2)†]U∗, |
where △′=(T2(T2)∗+(TS+SN)(In−t−(N2)†N2)(TS+SN)∗)−1.
Then it follows that
A2∈CEPn⟺A2(A2)†=(A2)†A2⟺(TS+SN)=(TS+SN)(N2)†N2, (N2)†N2=N2(N2)†⟺N2=0, TS+SN=0⟺S=0, N2=0. |
(d)⟹(a). We can get AA♢=AAD by A♢=A†,D. From (2.1), (2.4) and (2.5), AA♢=AAD is equivalent to
U[ΣKΣL00][(ΣK)†000]U∗=U[ΣKΣL00][(ΣK)D((ΣK)D)2ΣL00]U∗. |
Thus ΣK(ΣK)†=ΣK(ΣK)D. Then we have ΣK=(ΣK)2(ΣK)D which implies Ind(ΣK)≤1, moreover Ind(A)≤2.
Then let A be the form of (2.7). For Ind(A)≤2, we obtain N2=0. Representations (4.1) and (4.2) directly lead to
AA♢=AAD⟺U[TS0N][T∗△0PNS∗△0]U∗=U[TS0N][T−1(Tk+1)−1˜T00]U∗⟺[It0NPNS∗△0]U∗=U[It(Tk)−1˜T00]. |
Hence we get ˜T=0 which implies S=0.
(a)⟹(d). It can be directly checked.
(a)⟺(e). From the definition of A♢ and together with Lemma 2.3, it follows that
![]() |
From [7], it is shown that A♢= is equivalent to A♢=AD,† by using the Hartwig-Spindelböck decomposition. Now we can verify the equivalence of A♢=
and A♢=AD,† by Core-EP decomposition.
Theorem 4.4. Let A∈Cn×n be decomposed by (2.7). Then the following statements are equivalent:
(a) A♢= ;
(b) SN=0 and N2=0;
(c) A♢=AD,†.
Proof. (a)⟺(b). According to Corollary 3.3 in [9], we have that
Ak(Ak)†=U[It000]U∗. |
From the definition of A♢, and (2.9) together with the equation above, it follows that
![]() |
(b)⟺(c). From the definition of A♢ and AD,† together with (4.2), by using Lemma 2.3, it follows that
A♢=AD,†⟺A2A†=(AD,†)†⟺U[TSPN0NPN]U∗=(U[T−1(Tk+1)−1˜TPN00]U∗)†⟺SPN=0, NPN=0, ˜TPN=0⟺SN=0, N2=0, |
where ˜T=k−1∑j=0TjSNk−1−j.
Remark 4.5. If A of the form (2.7) is nilpotent, it follows that A=UNU∗. Then the (a) of Theorem 4.3 and the (b) of the Theorem 4.4 are equivalent to N2=0. In other words, if A is nilpotent, then it follows that the conditions A♢=AD,A♢= , A♢=AD,†,A♢=A†,D and A♢=
are equivalent.
In [13,Theorem 4], the author gave some equivalent conditions for A♢∈CEPn. Then we will give some necessary and sufficient conditions for A♢ which belongs to some special matrix classes by using Core-EP decomposition.
Theorem 4.6. Let A∈Cn×n be the form of (2.7). Then,
(a) A♢∈CCMn⟺N2=0;
(b) A♢∈CPn⟺N2=0 and T=TT∗+SPNS∗;
(c) A♢∈COPn⟺ T=It,SN=0 and N2=0 (or A2=A1. where A1 is presented in Lemma 2.2.)
Proof. (a). From the definition of BT-inverse, it follows that
A♢∈CCMn⟺(A2A†)†∈CCMn⟺A2A†∈CCMn. |
By (2.7) and (2.9), we obtain that
A2A†=U[TSPN0NPN]U∗. |
Thus A♢∈CCMn⟺N2N†=0⟺N2=0 which establishes point (a) of the theorem.
(b). For A♢∈CPn⊆CCMn, we have N2=0. From (4.1), now we have that
A♢=U[T∗△0PNS∗△0]U∗, |
where △=(TT∗+SPNS∗)−1.
Since A♢∈CPn, we get that T∗△=It, hence T=(△∗)−1=△−1. The sufficient condition of (b) can be directly checked, therefore point (b) of the theorem holds.
(c). It can be directly checked that A2=A1 is equivalent to T=It,SN=0 and N2=0 by Core-EP decomposition. For A♢∈COPn⊆CPn, we have N2=0 and T=△−1. From (4.1), we have
A♢=U[Ir0PNS∗△0]U∗, |
where △=(TT∗+SPNS∗)−1.
Since A♢∈COPn, we get that SPN=0 which implies T=It,SN=0. The sufficient condition of (c) can be directly checked, therefore point (c) of the theorem holds.
Remark 4.7. If A of the form (2.7) is nilpotent which implies A=UNU∗, then A♢∈CCMn or CPn or COPn is equivalent to A2=0 (or N2=0).
From [13], it is known that A♢A=AA♢ and (A♢)†=(A†)♢ are both satisfied when A∈CEPn, but we can't conclude A∈CEPn when A♢A=AA♢ or (A♢)†=(A†)♢ holds. How to establish an equivalence relation between them, the following theorem will give.
Theorem 4.8. Let A∈Cn×n written as in (2.1). Then the following statements are equivalent:
(a) A∈CEPn;
(b) AA♢=A♢A and A∈CCMn;
(c) (A♢)†=(A†)♢ and A∈CCMn;
(d) (A♢)m=(A†)m for some m≥2 and A∈CCMn.
Proof. That (a) implies items (b),(c) and (d) can be checked directly by the definition of A♢.
(b)⇒(a). For A∈CCMn, we get that K is nonsingular. By (2.5) and (2.6), we get that A♢= and A
=
A. Hence it follows that A∈CEPn by [3,Theorem 3].
(c)⇒(a). This follows similarly as in the part (b)⇒(a).
(d)⇒(a). It is known that A∈CEPn is equivalent to L=0. Combining (2.3), (2.5) with (A♢)m=(A†)m leads to L=0 which means A∈CEPn.
Finally, we study the representations for the BT-inverse. In [4], let A∈CCMn. While =A#AA† or (A2A†)†, the author gave new representations by the maximal matrix classes such as
=XAY or (A2Z)† where R(XA)⊆R(A) and Y∈A{1,3} or Z∈A{1,3}. Similarly, the author in [21] gave the representations of
, AD,† by the maximal classes. Now, we will derive the representations of BT-inverse by the maximal classes. We first give the important lemma as follows.
Lemma 5.1. [22] Let A,B,C∈Cn×n. Then the matrix equation AXB=C is consistent if and only if for some A(1)∈A{1}, B(1)∈B{1},
AA(1)CB(1)B=C, |
in which case the general solution is
X=A(1)CB(1)+Z−A(1)AZBB(1), |
for arbitrary Z∈Cn×n.
Theorem 5.2. Let A∈Cn×n of rank r has the form (2.1). Then the following conditions are equivalent:
(a) A♢=(A2X)†;
(b) A2X=APA;
(c) X=P(A2)†A†+(In−P(A2)†)Z, for arbitrary Z∈Cn×n;
(d) X can be expressed as
X=U[P∗R†ΣK+(Ir−P∗R†P)Z1−P∗R†QZ3(Ir−P∗R†P)Z2−P∗R†QZ4Q∗R†ΣK−Q∗R†PZ1+(In−r−Q∗R†Q)Z3−Q∗R†PZ2+(In−r−Q∗R†Q)Z4]U∗, |
where R=PP∗+QQ∗, P=(ΣK)2 and Q=ΣKΣL, for arbitrary Z1,Z2,Z3,Z4.
Proof. (a)⇒(b). Since A♢=(APA)†=(A2X)†, we have A2X=APA.
(b)⇒(c). It is evident that P(A2)†A† satisfies the equation
A2X=APA. | (5.1) |
Applying Lemma 5.1 to this equation, the general solution of (4.3) is given by
X=P(A2)†A†+(In−P(A2)†)Z, |
for arbitrary Z∈Cn×n.
(c)⟺(d). From (2.1), it follows that
A2=U[(ΣK)2ΣKΣL00]U∗, | (5.2) |
and applying [23,Lemma 1] to (5.2), we obtain that
(A2)†=U[P∗R†0Q∗R†0]U∗, |
where R=PP∗+QQ∗, P=(ΣK)2 and Q=ΣKΣL. Next, partitioning accordingly
Z=U[Z1Z2Z3Z4]U∗, |
a straightforward computation shows that X=P(A2)†A†+(In−P(A2)†)Z is equivalent to
X=U[P∗R†ΣK+(Ir−P∗R†P)Z1−P∗R†QZ3(Ir−P∗R†P)Z2−P∗R†QZ4Q∗R†ΣK−Q∗R†PZ1+(In−r−Q∗R†Q)Z3−Q∗R†PZ2+(In−r−Q∗R†Q)Z4]U∗, | (5.3) |
where R=PP∗+QQ∗, P=(ΣK)2 and Q=ΣKΣL, for arbitrary Z1,Z2,Z3,Z4.
(c)⇒(a). By a direct calculation, we have that A2X=A2A†. Therefore
(A2X)†=(A2A†)†=A♢. |
Theorem 5.3. Let A∈Cn×n be of the form (2.1), X,Y∈APA{1}. Then the following conditions are equivalent:
(a) A♢=XAPAY;
(b) XAPA=P(A2)† and APAY=A(APA)†;
(c) X=(APA)†+Z(In−PAPA) and Y=(APA)†+(In−P(APA)†)W, for arbitrary Z,W∈Cn×n;
(d) X,Y can be expressed as
X=U[(ΣK)†+Z1(Ir−ΣK(ΣK)†)Z2Z3(Ir−ΣK(ΣK)†)Z4]U∗, |
for arbitrary Z1,Z2,Z3,Z4;
Y=U[(ΣK)†+(Ir−(ΣK)†ΣK)W1(Ir−(ΣK)†ΣK)W2W3W4]U∗, |
for arbitrary W1,W2,W3,W4.
Proof. (a)⇒(b). Postmultiplying A♢=XAPAY by APA. For Y∈APA{1}, it follows that XAPA=P(A2)†. Premultiplying A♢=XAPAY by APA. Since X∈APA{1}, it follows that APAY=APAA♢=AA♢.
(b)⇒(c). Applying Lemma 5.1 to two equations XAPA=(APA)†APA and APAY=A(APA)† respectively, the general solutions are given by X=(APA)†+Z(In−PAPA) for arbitrary Z∈Cn×n and Y=(APA)†+(In−P(APA)†)W for arbitrary W∈Cn×n.
(c)⇒(d). Assume that A has the form given in (2.1), we have
In−PAPA=U[Ir−ΣK(ΣK)†00In−r]U∗, |
In−P(APA)†=U[Ir−(ΣK)†ΣK00In−r]U∗. |
Next, partitioning accordingly
Z=U[Z1Z2Z3Z4]U∗,W=U[W1W2W3W4]U∗, |
a straightforward shows that X=(APA)†+Z(In−PAPA) is equivalent to
X=U[(ΣK)†+Z1(Ir−ΣK(ΣK)†)Z2Z3(Ir−ΣK(ΣK)†)Z4]U∗, | (5.4) |
for arbitrary Z1,Z2,Z3,Z4. Y=(APA)†+(In−P(APA)†)W is equivalent to
Y=U[(ΣK)†+(Ir−(ΣK)†ΣK)W1(Ir−(ΣK)†ΣK)W2W3W4]U∗, | (5.5) |
for arbitrary W1,W2,W3,W4.
(d)⇒(a). According to (5.4) and (5.5), a straightforward computation shows that
XAPAY=U[(ΣK)†+Z1(Ir−ΣK(ΣK)†)Z2Z3(Ir−ΣK(ΣK)†)Z4][ΣK(ΣK)†000]U∗=U[(ΣK)†000]U∗=A♢. |
In this work, different characteristics of the BT-inverse of a square matrix have been developed. Some necessary and sufficient conditions for a matrix to be the BT-inverse have been derived. The Core-EP decomposition is efficient for investigating the relationships between the BT-inverse and other generalized inverses. The expression of BT-inverse has been extended to more general ones by the maximal classes of matrices.
The authors are thankful to four anonymous referees for their careful reading, detailed corrections, insightful comments and pertinent suggestions on the first version of the paper, which enhance the presentation of the results distinctly. This research is supported by the Natural Science Foundation of China under Grants 11961076.
The authors declare no conflict of interest.
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