vol18_pp117-125. 106KB Jun 04 2011 12:06:02 AM
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 117-125, February 2009
ELA
http://math.technion.ac.il/iic/ela
SOME RESULTS ON GROUP INVERSES OF BLOCK MATRICES
OVER SKEW FIELDS∗
CHANGJIANG BU† , JIEMEI ZHAO† , AND KUIZE ZHANG†
Abstract. In this paper, necessary and
sufficient conditions are given for the existence of
A A over any skew field, where A, B are both square and
the group inverse of the block matrix B
0
rank(B) ≥ rank(A). The representation of this group inverse and some relative additive results are
also given.
Key words. Skew, Block matrix, Group inverse.
AMS subject classifications. 15A09.
1. Introduction. Let K be a skew field and K n×n be the set of all matrices
over K. For A ∈ K n×n , the matrix X ∈ K n×n is said to be the group inverse of A, if
AXA = A, XAX = X, AX = XA.
♯
We then write X = A . It is well known that if A♯ exists, it is unique; see [16].
Research on representations of the group inverse of block matrices is an important
effort in generalized inverse theory of matrices; see [14] and [13]. Indeed, generalized
inverses are useful tools in areas such as special matrix theory, singular differential
and difference equations and graph theory; see [5], [9] [11], [12] and [15]. For example,
in [9] it is shown that the adjacency matrix of a bipartite graph can be written in the
0 B
, and necessary and sufficient conditions are given for the existence
form of
C 0
0 B
.
and representation of the group inverse of a block matrix
C 0
In 1979, Campbell and Meyer proposed the problem of finding
repre an explicit
A B
in terms
sentation for the Drazin (group) inverse of a 2 × 2 block matrix
C D
of its sub-blocks, where A and D are required to be square matrices; see [5]. In [10] a
A B
is given under the ascondition for the existence of the group inverse of
C D
∗ Received by the editors October 11, 2008. Accepted for publication February 20, 2009. Handling
Editor: Ravindra B. Bapat.
† Department of Applied Mathematics, College of Science, Harbin Engineering University, Harbin
150001, P. R. China (buchangjiang@hrbeu.edu.cn, zhaojiemei500@163.com, zkz0017@163.com).
Supported by the Natural Science Foundation of Heilongjiang Province, No.159110120002.
117
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 117-125, February 2009
ELA
http://math.technion.ac.il/iic/ela
118
Changjiang Bu, Jiemei Zhao, and Kuize Zhang
sumption that A and (I +CA−2 B) are both invertible over any field; however, the representation of thegroup inverse
is not given. The representation of the group inverse
A B
over skew fields has been given in 2001; see [6]. The repof a block matrix
0 C
A B
(A
resentation of the Drazin (group) inverse of a block matrix of the form
C 0
is square, 0 is square null matrix) has not been given since it was proposed as a problem by Campbell in 1983; see [4]. However, there are some references in the literature
A B
about representations of the Drazin (group) inverse of the block matrices
C 0
under certain conditions. Some results are on matrices over the field of complex numbers, e.g., in [8]; or when A = B = In in [7]; or when A, B, C ∈ {P, P ∗ , P P ∗ }, P 2 = P
and P ∗ is the conjugate transpose of P . Some results are over skew fields, e.g., in [1],
when A = In and rank(CB)2 = rank(B) = rank(C); in [3] when A = B, A2 = A. In
addition, in [2] results are given on the group inverse of the product of two matrices
over a skew field, as well as some related properties.
In this paper, we mainly give necessary and sufficient conditions for the
exisA A
or
tence and the representation of the group inverse of a block matrix
B 0
A B
, where A, B ∈ K n×n , rank(B) ≥ rank(A). We also give a sufficient
A 0
condition for AB to be similar to BA.
Letting A ∈ K m×n , the order of the maximum invertible sub-block of A is said
to be the rank of A, denoted by rank(A); see [17]. Let A, B ∈ K n×n . If there is an
invertible matrix P ∈ K n×n such that B = P AP −1 , then A and B are similar; see
[17].
2. Some Lemmas.
Lemma 2.1. Let A, B ∈ K n×n . If rank(A) = r, rank(B) = rank(AB) =
rank(BA), then there are invertible matrices P, Q ∈ K n×n such that
A=P
Ir
0
0
0
Q, B = Q
−1
B1
Y B1
B1 X
Y B1 X
P −1 ,
where B1 ∈ K r×r , X ∈ K r×(n−r), and Y ∈ K (n−r)×r .
Proof. Since rank(A) = r, there are nonsingular matrices P, Q ∈ K n×n such that
A=P
Ir
0
0
0
Q, B = Q−1
B1
B3
B2
B4
P −1 ,
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 117-125, February 2009
ELA
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Group Inverses of Block Matrices over Skew Fields
119
where B1 ∈ K r×r , B2 ∈ K r×(n−r) , B3 ∈ K (n−r)×r , and B4 ∈ K (n−r)×(n−r). From
rank(B) = rank(AB), we have
B3 = Y B1 , B4 = Y B2 , Y ∈ K (n−r)×r .
Since rank(B) = rank(BA), we obtain
B2 = B1 X, B4 = B3 X, X ∈ K r×(n−r) .
So
B = Q−1
B1
Y B1
B1 X
Y B1 X
P −1 .
A 0
∈ K n×n .
B 0
Then the group inverse
of M exists if and only if the group inverse of A exists and
A
. If the group inverse of M exists, then
rank(A) = rank
B
Lemma 2.2. [6] Let A ∈ K r×r , B ∈ K (n−r)×r , M =
♯
M =
A♯
B(A♯ )2
0
0
.
A B
∈ K n×n .
Lemma 2.3. [6] Let A ∈ K
, B ∈ K
, M =
0 0
Then the group inverse of M exists if and only if the group inverse of A exists and
rank(A) = rank A B . If the group inverse of M exists, then
r×r
M♯ =
r×(n−r)
A♯
0
(A♯ )2 B
0
.
Lemma 2.4. [2] Let A ∈ K m×n , B ∈ K n×m . If rank(A) = rank(BA), rank(B)
= rank(AB), then the group inverse of AB and BA exist.
Lemma 2.5. Let A, B ∈ K n×n . If rank(A) = rank(B) = rank(AB) =
rank(BA), then the following conclusions hold:
(i)
(ii)
(iii)
(iv)
(v)
AB(AB)♯ A = A,
A(BA)♯ BA = A,
BA(BA)♯ B = B,
B(AB)♯ A = BA(BA)♯ ,
A(BA)♯ = (AB)♯ A.
Electronic Journal of Linear Algebra ISSN 1081-3810
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Volume 18, pp. 117-125, February 2009
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120
Changjiang Bu, Jiemei Zhao, and Kuize Zhang
Proof. Suppose rank(A) = r. By Lemma 2.1, we have
A=P
Ir
0
0
0
Q, B = Q
−1
B1
Y B1
B1 X
Y B1 X
P −1 ,
where B1 ∈ K r×r , X ∈ K r×(n−r) , Y ∈ K (n−r)×r . Then
AB = P
B1
0
B1 X
0
P −1 , BA = Q−1
B1
Y B1
0
0
Q.
Since rank(A) = rank(B), we have that B1 is invertible. By using Lemma 2.2 and
Lemma 2.3, we get
(AB)♯ = P
B1 −1
0
B1 −1 X
0
P −1 , (BA)♯ = Q−1
B1 −1
Y B1 −1
0
0
Q.
Then
(ii)
(iii)
(iv)
(v)
Ir 0
Q = A,
0 0
Ir 0
Q = A,
A(BA)♯ BA = P
0 0
B1
B1 X
♯
−1
P −1 = B,
BA(BA) B = Q
Y B1 Y B1 X
Ir 0
♯
−1
B(AB) A = Q
Q = BA(BA)♯ ,
Y 0
B1 −1 0
♯
Q = (AB)♯ A.
A(BA) = P
0
0
(i) AB(AB)♯ A = P
3. Conclusions.
Theorem 3.1. Let M =
A
B
A
0
, where A, B ∈ K n×n , rank(B) ≥ rank(A)
= r. Then
(i) The group inverse of M exists if and only if rank(A) = rank(B) = rank(AB) =
rank(BA).
M11 M12
, where
(ii) If the group inverse of M exists, then M ♯ =
M21 M22
M11 = (AB)♯ A − (AB)♯ A2 (BA)♯ B,
M12 = (AB)♯ A,
M21 = (BA)♯ B − B(AB)♯ A2 (BA)♯ + B(AB)♯ A(AB)♯ A2 (BA)♯ B,
M22 = −B(AB)♯ A2 (BA)♯ .
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 117-125, February 2009
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121
Group Inverses of Block Matrices over Skew Fields
Proof. (i) It is obvious that the condition is sufficient. Now we show that the
condition is necessary.
rank(M ) = rank
2
A
B
rank(M ) = rank
A
0
= rank
A2 + AB
BA
0
B
A2
BA
A
0
= rank(A) + rank(B),
= rank
AB
0
A2
BA
.
Since the group inverse of M exists if and only if rank(M ) = rank(M 2 ), we have
rank(A) + rank(B) = rank(M 2 )
≤ rank(AB) + rank
A2
BA
≤ rank(AB) + rank(A),
rank(A) + rank(B) = rank(M 2 )
≤ rank AB
A2
+ rank(BA)
≤ rank(BA) + rank(A).
Then rank(B) ≤ rank(AB), and rank(B) ≤ rank(BA). Therefore
rank(B) = rank(AB) = rank(BA).
From rank(B) = rank(AB) ≤ rank(A), and rank(A) ≤ rank(B), we have
rank(A) = rank(B).
Since rank(A)+rank(B) ≤ rank AB A2 +rank(BA), and rank AB
rank(A), we get rank AB A2 = rank(A). Thus
rank
AB
A2
= rank(AB).
Then there exists a matrix U ∈ K n×n such that ABU = A2 . Then
AB
0
= rank(AB) + rank(BA).
rank(M 2 ) = rank
0
BA
So we get
rank(A) = rank(B) = rank(AB) = rank(BA).
A2
≤
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 117-125, February 2009
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122
Changjiang Bu, Jiemei Zhao, and Kuize Zhang
M11 M12
. We will prove that the matrix X satisfies the
M21 M22
conditions of the group inverse. Firstly, we compute
(ii) Let X =
MX =
AM11 + AM21
BM11
XM =
AM12 + AM22
BM12
M11 A + M12 B
M21 A + M22 B
M11 A
M21 A
,
.
Applying Lemma 2.5 (i), (ii), and (v), we have
AM11 + AM21 = A(AB)♯ A − A(AB)♯ A2 (BA)♯ B + A(BA)♯ B − AB(AB)♯ A2 (BA)♯
+ AB(AB)♯ A(AB)♯ A2 (BA)♯ B
= A(BA)♯ B,
M11 A + M12 B = (AB)♯ A2 − (AB)♯ A2 (BA)♯ BA + (AB)♯ AB
= (AB)♯ A2 − (AB)♯ A2 + (AB)♯ AB
= A(BA)♯ B.
Using Lemma 2.5 (i), (ii), and (v), we get
AM12 + AM22 = A(AB)♯ A − AB(AB)♯ A2 (BA)♯
= A(AB)♯ A − A2 (BA)♯
= 0,
M11 A = (AB)♯ A2 − (AB)♯ A2 (BA)♯ BA
= (AB)♯ A2 − (AB)♯ A2
= 0.
From Lemma 2.5 (ii), we obtain
BM11 = B(AB)♯ A − B(AB)♯ A2 (BA)♯ B,
M21 A + M22 B = (BA)♯ BA − B(AB)♯ A2 (BA)♯ A + B(AB)♯ A2 (BA)♯ A(BA)♯ BA
− B(AB)♯ A2 (BA)♯ B
= (BA)♯ BA − [B(AB)♯ A2 (BA)♯ A − B(AB)♯ A2 (BA)♯ A]
− B(AB)♯ A2 (BA)♯ B
= B(AB)♯ A − B(AB)♯ A2 (BA)♯ B.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 117-125, February 2009
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123
Group Inverses of Block Matrices over Skew Fields
Using Lemma 2.5 (ii), we have
BM12 = B(AB)♯ A,
M21 A = (BA)♯ BA − B(AB)♯ A2 (BA)♯ A + B(AB)♯ A2 (BA)♯ A(BA)♯ BA
= B(AB)♯ A.
So
M X = XM =
A(BA)♯ B
♯
B(AB) A − B(AB)♯ A2 (BA)♯ B
0
B(AB)♯ A
.
Secondly,
=
M XM
=
A(BA)♯ B
A A
♯
B 0
B(AB) A − B(AB)♯ A2 (BA)♯ B
X11 X12
.
X21
0
0
B(AB)♯ A
Applying Lemma 2.5 (i) and (iii), we compute
X11 = A2 (BA)♯ B + AB(AB)♯ A − AB(AB)♯ A2 (BA)♯ B
= AB(AB)♯ A
= A,
X12 = AB(AB)♯ A = A,
X21 = BA(BA)♯ B = B.
Hence
M XM =
A
B
A
0
.
Finally,
XM X
=
=
M11 M12
A(BA)♯ B
M
M22
B(AB)♯ A − B(AB)♯ A2 (BA)♯ B
21
Y11 Y12
.
Y21 Y22
0
B(AB)♯ A
Then
Y11 = (AB)♯ A2 (BA)♯ B − (AB)♯ A2 (BA)♯ BA(BA)♯ B + (AB)♯ AB(AB)♯ A
− (AB)♯ AB(AB)♯ A2 (BA)♯ B
= (AB)♯ A − (AB)♯ A2 (BA)♯ B
= M11 ,
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 117-125, February 2009
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124
Changjiang Bu, Jiemei Zhao, and Kuize Zhang
and
=
=
=
=
Y12
M12 B(AB)♯ A
(AB)♯ AB(AB)♯ A
(AB)♯ A
M12 .
We can easily get
Y21 = M21 A(BA)♯ B + M22 B(AB)♯ A − M22 B(AB)♯ A2 (BA)♯ B
= M21 ;
Y22 = M22 B(AB)♯ A = M22 .
So we have X = M ♯ .
Theorem 3.2. Let M =
A
A
B
0
, where A, B ∈ K n×n , rank(B) ≥ rank(A)
= r. Then
(i) The group inverse of M exists if and only if rank(A) = rank(B) = rank(AB) =
rank(BA).
Z11 Z12
♯
, where
(ii) If the group inverse of M exists, then M =
Z21 Z22
Z11 = (AB)♯ A − B(AB)♯ A2 (BA)♯ ,
Z12 = B(AB)♯ − (AB)♯ A2 (BA)♯ B + B(AB)♯ A2 (BA)♯ A(BA)♯ B,
Z21 = (AB)♯ A,
Z22 = − (AB)♯ A2 (BA)♯ B.
Proof. Let X =
M11
M21
M X = XM =
M12
M22
. By Lemma 2.5, we have
B(AB)♯ A A(BA)♯ B − B(AB)♯ A2 (BA)♯ B
0
A(BA)♯ B
.
Furthermore, we can prove M XM = M, XM X = X easily. Thus, X = M ♯ .
Theorem 3.3. Let A, B ∈ K n×n , if rank(B) = rank(AB) = rank(BA). Then
AB and BA are similar.
Proof. Suppose rank(A) = r, using Lemma 2.1, there are invertible matrices
P, Q ∈ K n×n such that
Ir 0
B1
B1 X
A=P
P −1 ,
Q, B = Q−1
Y B1 Y B1 X
0 0
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 117-125, February 2009
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125
Group Inverses of Block Matrices over Skew Fields
where B1 ∈ K r×r , X ∈ K r×(n−r) , Y ∈ K (n−r)×r . Hence
B1 B1 X
AB = P
P −1
0
0
Ir −X
Ir
B1 0
= P
0 In−r
0
0 0
BA
In−r
P −1 ,
B1 0
= Q
Q
Y B1 0
Ir
B1
0
= Q−1
Y In−r
0
−1
X
0
0
Ir
−Y
0
In−r
Q.
So AB and BA are similar.
REFERENCES
[1] C. Bu. On group inverses of block matrices over skew fields. J. Math., 35:49–52, 2002.
[2] C. Bu and C. Cao. Group inverse of product of two matrices over skew field. J. Math. Study,
35:435–438, 2002.
[3] C. Bu, J. Zhao, and J. Zheng. Group inverse for a class 2 × 2 block matrices over skew fields.
Appl. Math. Comput., 204:45–49, 2008.
[4] S.L. Campbell. The Drazin inverse and systems of second order linear differential equations.
Linear Multilinear Algebra, 14:195–198, 1983.
[5] S.L. Campbell and C.D. Meyer. Generalized Inverses of Linear Transformations. Dover, New
York, 1991 (Originally published: Pitman, London, 1979).
[6] C. Cao. Some results of group inverses for partitioned matrices over skew fields. Heilongjiang
Daxue Ziran Kexue Xuebao (Chinese), 18:5–7, 2001.
[7] C. Cao and X. Tang. Representations of the group inverse of some 2 × 2 block matrices. Int.
Math. Forum, 31:1511–1517, 2006.
[8] N. Castro-Gonz´
ales and E. Dopazo. Representations of the Drazin inverse for a class of block
matrices. Linear Algebra Appl., 400:253–269, 2005.
[9] M. Catral, D.D. Olesky, and P. van den Driessche. Group inverses of matrices with path graphs.
Electron. J. Linear Algebra, 17:219–233, 2008.
[10] X. Chen and R.E. Hartwig. The group inverse of a triangular matrix. Linear Algebra Appl.,
237/238:97–108, 1996.
[11] G.H. Golub and C. Greif. On solving block-structured indefinite linear systems. SIAM J. Sci.
Comput., 24:2076–2092, 2003.
[12] I.C.F. Ipsen. A note on preconditioning nonsymmetric matrices. SIAM J. Sci. Comput.,
23:1050–1051, 2001.
[13] K.P.S. Bhaskara Rao. The Theory of Generalized Inverses over Commutative Rings. TaylorFrancis, New York, 2002.
[14] G. Wang, Y. Wei, and S. Qian. Generalized Inverse Theory and Computations. Science Press,
Beijing, 2003.
[15] Y. Wei and H. Diao. On group inverse of singular Toeplitz matrices. Linear Algebra Appl.,
399:109–123, 2005.
[16] Wa-Jin Zhuang. Involutory functions and generalized inverses of matrices over an arbitrary
skew fields. Northeast. Math. J., (Chinese) 1:57–65, 1987.
[17] W. Zhuang. The Guidance of Matrix Theory over Skew Fields (Chinese). Science Press, Beijing,
2006.
A publication of the International Linear Algebra Society
Volume 18, pp. 117-125, February 2009
ELA
http://math.technion.ac.il/iic/ela
SOME RESULTS ON GROUP INVERSES OF BLOCK MATRICES
OVER SKEW FIELDS∗
CHANGJIANG BU† , JIEMEI ZHAO† , AND KUIZE ZHANG†
Abstract. In this paper, necessary and
sufficient conditions are given for the existence of
A A over any skew field, where A, B are both square and
the group inverse of the block matrix B
0
rank(B) ≥ rank(A). The representation of this group inverse and some relative additive results are
also given.
Key words. Skew, Block matrix, Group inverse.
AMS subject classifications. 15A09.
1. Introduction. Let K be a skew field and K n×n be the set of all matrices
over K. For A ∈ K n×n , the matrix X ∈ K n×n is said to be the group inverse of A, if
AXA = A, XAX = X, AX = XA.
♯
We then write X = A . It is well known that if A♯ exists, it is unique; see [16].
Research on representations of the group inverse of block matrices is an important
effort in generalized inverse theory of matrices; see [14] and [13]. Indeed, generalized
inverses are useful tools in areas such as special matrix theory, singular differential
and difference equations and graph theory; see [5], [9] [11], [12] and [15]. For example,
in [9] it is shown that the adjacency matrix of a bipartite graph can be written in the
0 B
, and necessary and sufficient conditions are given for the existence
form of
C 0
0 B
.
and representation of the group inverse of a block matrix
C 0
In 1979, Campbell and Meyer proposed the problem of finding
repre an explicit
A B
in terms
sentation for the Drazin (group) inverse of a 2 × 2 block matrix
C D
of its sub-blocks, where A and D are required to be square matrices; see [5]. In [10] a
A B
is given under the ascondition for the existence of the group inverse of
C D
∗ Received by the editors October 11, 2008. Accepted for publication February 20, 2009. Handling
Editor: Ravindra B. Bapat.
† Department of Applied Mathematics, College of Science, Harbin Engineering University, Harbin
150001, P. R. China (buchangjiang@hrbeu.edu.cn, zhaojiemei500@163.com, zkz0017@163.com).
Supported by the Natural Science Foundation of Heilongjiang Province, No.159110120002.
117
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 117-125, February 2009
ELA
http://math.technion.ac.il/iic/ela
118
Changjiang Bu, Jiemei Zhao, and Kuize Zhang
sumption that A and (I +CA−2 B) are both invertible over any field; however, the representation of thegroup inverse
is not given. The representation of the group inverse
A B
over skew fields has been given in 2001; see [6]. The repof a block matrix
0 C
A B
(A
resentation of the Drazin (group) inverse of a block matrix of the form
C 0
is square, 0 is square null matrix) has not been given since it was proposed as a problem by Campbell in 1983; see [4]. However, there are some references in the literature
A B
about representations of the Drazin (group) inverse of the block matrices
C 0
under certain conditions. Some results are on matrices over the field of complex numbers, e.g., in [8]; or when A = B = In in [7]; or when A, B, C ∈ {P, P ∗ , P P ∗ }, P 2 = P
and P ∗ is the conjugate transpose of P . Some results are over skew fields, e.g., in [1],
when A = In and rank(CB)2 = rank(B) = rank(C); in [3] when A = B, A2 = A. In
addition, in [2] results are given on the group inverse of the product of two matrices
over a skew field, as well as some related properties.
In this paper, we mainly give necessary and sufficient conditions for the
exisA A
or
tence and the representation of the group inverse of a block matrix
B 0
A B
, where A, B ∈ K n×n , rank(B) ≥ rank(A). We also give a sufficient
A 0
condition for AB to be similar to BA.
Letting A ∈ K m×n , the order of the maximum invertible sub-block of A is said
to be the rank of A, denoted by rank(A); see [17]. Let A, B ∈ K n×n . If there is an
invertible matrix P ∈ K n×n such that B = P AP −1 , then A and B are similar; see
[17].
2. Some Lemmas.
Lemma 2.1. Let A, B ∈ K n×n . If rank(A) = r, rank(B) = rank(AB) =
rank(BA), then there are invertible matrices P, Q ∈ K n×n such that
A=P
Ir
0
0
0
Q, B = Q
−1
B1
Y B1
B1 X
Y B1 X
P −1 ,
where B1 ∈ K r×r , X ∈ K r×(n−r), and Y ∈ K (n−r)×r .
Proof. Since rank(A) = r, there are nonsingular matrices P, Q ∈ K n×n such that
A=P
Ir
0
0
0
Q, B = Q−1
B1
B3
B2
B4
P −1 ,
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 117-125, February 2009
ELA
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Group Inverses of Block Matrices over Skew Fields
119
where B1 ∈ K r×r , B2 ∈ K r×(n−r) , B3 ∈ K (n−r)×r , and B4 ∈ K (n−r)×(n−r). From
rank(B) = rank(AB), we have
B3 = Y B1 , B4 = Y B2 , Y ∈ K (n−r)×r .
Since rank(B) = rank(BA), we obtain
B2 = B1 X, B4 = B3 X, X ∈ K r×(n−r) .
So
B = Q−1
B1
Y B1
B1 X
Y B1 X
P −1 .
A 0
∈ K n×n .
B 0
Then the group inverse
of M exists if and only if the group inverse of A exists and
A
. If the group inverse of M exists, then
rank(A) = rank
B
Lemma 2.2. [6] Let A ∈ K r×r , B ∈ K (n−r)×r , M =
♯
M =
A♯
B(A♯ )2
0
0
.
A B
∈ K n×n .
Lemma 2.3. [6] Let A ∈ K
, B ∈ K
, M =
0 0
Then the group inverse of M exists if and only if the group inverse of A exists and
rank(A) = rank A B . If the group inverse of M exists, then
r×r
M♯ =
r×(n−r)
A♯
0
(A♯ )2 B
0
.
Lemma 2.4. [2] Let A ∈ K m×n , B ∈ K n×m . If rank(A) = rank(BA), rank(B)
= rank(AB), then the group inverse of AB and BA exist.
Lemma 2.5. Let A, B ∈ K n×n . If rank(A) = rank(B) = rank(AB) =
rank(BA), then the following conclusions hold:
(i)
(ii)
(iii)
(iv)
(v)
AB(AB)♯ A = A,
A(BA)♯ BA = A,
BA(BA)♯ B = B,
B(AB)♯ A = BA(BA)♯ ,
A(BA)♯ = (AB)♯ A.
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Changjiang Bu, Jiemei Zhao, and Kuize Zhang
Proof. Suppose rank(A) = r. By Lemma 2.1, we have
A=P
Ir
0
0
0
Q, B = Q
−1
B1
Y B1
B1 X
Y B1 X
P −1 ,
where B1 ∈ K r×r , X ∈ K r×(n−r) , Y ∈ K (n−r)×r . Then
AB = P
B1
0
B1 X
0
P −1 , BA = Q−1
B1
Y B1
0
0
Q.
Since rank(A) = rank(B), we have that B1 is invertible. By using Lemma 2.2 and
Lemma 2.3, we get
(AB)♯ = P
B1 −1
0
B1 −1 X
0
P −1 , (BA)♯ = Q−1
B1 −1
Y B1 −1
0
0
Q.
Then
(ii)
(iii)
(iv)
(v)
Ir 0
Q = A,
0 0
Ir 0
Q = A,
A(BA)♯ BA = P
0 0
B1
B1 X
♯
−1
P −1 = B,
BA(BA) B = Q
Y B1 Y B1 X
Ir 0
♯
−1
B(AB) A = Q
Q = BA(BA)♯ ,
Y 0
B1 −1 0
♯
Q = (AB)♯ A.
A(BA) = P
0
0
(i) AB(AB)♯ A = P
3. Conclusions.
Theorem 3.1. Let M =
A
B
A
0
, where A, B ∈ K n×n , rank(B) ≥ rank(A)
= r. Then
(i) The group inverse of M exists if and only if rank(A) = rank(B) = rank(AB) =
rank(BA).
M11 M12
, where
(ii) If the group inverse of M exists, then M ♯ =
M21 M22
M11 = (AB)♯ A − (AB)♯ A2 (BA)♯ B,
M12 = (AB)♯ A,
M21 = (BA)♯ B − B(AB)♯ A2 (BA)♯ + B(AB)♯ A(AB)♯ A2 (BA)♯ B,
M22 = −B(AB)♯ A2 (BA)♯ .
Electronic Journal of Linear Algebra ISSN 1081-3810
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121
Group Inverses of Block Matrices over Skew Fields
Proof. (i) It is obvious that the condition is sufficient. Now we show that the
condition is necessary.
rank(M ) = rank
2
A
B
rank(M ) = rank
A
0
= rank
A2 + AB
BA
0
B
A2
BA
A
0
= rank(A) + rank(B),
= rank
AB
0
A2
BA
.
Since the group inverse of M exists if and only if rank(M ) = rank(M 2 ), we have
rank(A) + rank(B) = rank(M 2 )
≤ rank(AB) + rank
A2
BA
≤ rank(AB) + rank(A),
rank(A) + rank(B) = rank(M 2 )
≤ rank AB
A2
+ rank(BA)
≤ rank(BA) + rank(A).
Then rank(B) ≤ rank(AB), and rank(B) ≤ rank(BA). Therefore
rank(B) = rank(AB) = rank(BA).
From rank(B) = rank(AB) ≤ rank(A), and rank(A) ≤ rank(B), we have
rank(A) = rank(B).
Since rank(A)+rank(B) ≤ rank AB A2 +rank(BA), and rank AB
rank(A), we get rank AB A2 = rank(A). Thus
rank
AB
A2
= rank(AB).
Then there exists a matrix U ∈ K n×n such that ABU = A2 . Then
AB
0
= rank(AB) + rank(BA).
rank(M 2 ) = rank
0
BA
So we get
rank(A) = rank(B) = rank(AB) = rank(BA).
A2
≤
Electronic Journal of Linear Algebra ISSN 1081-3810
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Changjiang Bu, Jiemei Zhao, and Kuize Zhang
M11 M12
. We will prove that the matrix X satisfies the
M21 M22
conditions of the group inverse. Firstly, we compute
(ii) Let X =
MX =
AM11 + AM21
BM11
XM =
AM12 + AM22
BM12
M11 A + M12 B
M21 A + M22 B
M11 A
M21 A
,
.
Applying Lemma 2.5 (i), (ii), and (v), we have
AM11 + AM21 = A(AB)♯ A − A(AB)♯ A2 (BA)♯ B + A(BA)♯ B − AB(AB)♯ A2 (BA)♯
+ AB(AB)♯ A(AB)♯ A2 (BA)♯ B
= A(BA)♯ B,
M11 A + M12 B = (AB)♯ A2 − (AB)♯ A2 (BA)♯ BA + (AB)♯ AB
= (AB)♯ A2 − (AB)♯ A2 + (AB)♯ AB
= A(BA)♯ B.
Using Lemma 2.5 (i), (ii), and (v), we get
AM12 + AM22 = A(AB)♯ A − AB(AB)♯ A2 (BA)♯
= A(AB)♯ A − A2 (BA)♯
= 0,
M11 A = (AB)♯ A2 − (AB)♯ A2 (BA)♯ BA
= (AB)♯ A2 − (AB)♯ A2
= 0.
From Lemma 2.5 (ii), we obtain
BM11 = B(AB)♯ A − B(AB)♯ A2 (BA)♯ B,
M21 A + M22 B = (BA)♯ BA − B(AB)♯ A2 (BA)♯ A + B(AB)♯ A2 (BA)♯ A(BA)♯ BA
− B(AB)♯ A2 (BA)♯ B
= (BA)♯ BA − [B(AB)♯ A2 (BA)♯ A − B(AB)♯ A2 (BA)♯ A]
− B(AB)♯ A2 (BA)♯ B
= B(AB)♯ A − B(AB)♯ A2 (BA)♯ B.
Electronic Journal of Linear Algebra ISSN 1081-3810
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Group Inverses of Block Matrices over Skew Fields
Using Lemma 2.5 (ii), we have
BM12 = B(AB)♯ A,
M21 A = (BA)♯ BA − B(AB)♯ A2 (BA)♯ A + B(AB)♯ A2 (BA)♯ A(BA)♯ BA
= B(AB)♯ A.
So
M X = XM =
A(BA)♯ B
♯
B(AB) A − B(AB)♯ A2 (BA)♯ B
0
B(AB)♯ A
.
Secondly,
=
M XM
=
A(BA)♯ B
A A
♯
B 0
B(AB) A − B(AB)♯ A2 (BA)♯ B
X11 X12
.
X21
0
0
B(AB)♯ A
Applying Lemma 2.5 (i) and (iii), we compute
X11 = A2 (BA)♯ B + AB(AB)♯ A − AB(AB)♯ A2 (BA)♯ B
= AB(AB)♯ A
= A,
X12 = AB(AB)♯ A = A,
X21 = BA(BA)♯ B = B.
Hence
M XM =
A
B
A
0
.
Finally,
XM X
=
=
M11 M12
A(BA)♯ B
M
M22
B(AB)♯ A − B(AB)♯ A2 (BA)♯ B
21
Y11 Y12
.
Y21 Y22
0
B(AB)♯ A
Then
Y11 = (AB)♯ A2 (BA)♯ B − (AB)♯ A2 (BA)♯ BA(BA)♯ B + (AB)♯ AB(AB)♯ A
− (AB)♯ AB(AB)♯ A2 (BA)♯ B
= (AB)♯ A − (AB)♯ A2 (BA)♯ B
= M11 ,
Electronic Journal of Linear Algebra ISSN 1081-3810
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Changjiang Bu, Jiemei Zhao, and Kuize Zhang
and
=
=
=
=
Y12
M12 B(AB)♯ A
(AB)♯ AB(AB)♯ A
(AB)♯ A
M12 .
We can easily get
Y21 = M21 A(BA)♯ B + M22 B(AB)♯ A − M22 B(AB)♯ A2 (BA)♯ B
= M21 ;
Y22 = M22 B(AB)♯ A = M22 .
So we have X = M ♯ .
Theorem 3.2. Let M =
A
A
B
0
, where A, B ∈ K n×n , rank(B) ≥ rank(A)
= r. Then
(i) The group inverse of M exists if and only if rank(A) = rank(B) = rank(AB) =
rank(BA).
Z11 Z12
♯
, where
(ii) If the group inverse of M exists, then M =
Z21 Z22
Z11 = (AB)♯ A − B(AB)♯ A2 (BA)♯ ,
Z12 = B(AB)♯ − (AB)♯ A2 (BA)♯ B + B(AB)♯ A2 (BA)♯ A(BA)♯ B,
Z21 = (AB)♯ A,
Z22 = − (AB)♯ A2 (BA)♯ B.
Proof. Let X =
M11
M21
M X = XM =
M12
M22
. By Lemma 2.5, we have
B(AB)♯ A A(BA)♯ B − B(AB)♯ A2 (BA)♯ B
0
A(BA)♯ B
.
Furthermore, we can prove M XM = M, XM X = X easily. Thus, X = M ♯ .
Theorem 3.3. Let A, B ∈ K n×n , if rank(B) = rank(AB) = rank(BA). Then
AB and BA are similar.
Proof. Suppose rank(A) = r, using Lemma 2.1, there are invertible matrices
P, Q ∈ K n×n such that
Ir 0
B1
B1 X
A=P
P −1 ,
Q, B = Q−1
Y B1 Y B1 X
0 0
Electronic Journal of Linear Algebra ISSN 1081-3810
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Group Inverses of Block Matrices over Skew Fields
where B1 ∈ K r×r , X ∈ K r×(n−r) , Y ∈ K (n−r)×r . Hence
B1 B1 X
AB = P
P −1
0
0
Ir −X
Ir
B1 0
= P
0 In−r
0
0 0
BA
In−r
P −1 ,
B1 0
= Q
Q
Y B1 0
Ir
B1
0
= Q−1
Y In−r
0
−1
X
0
0
Ir
−Y
0
In−r
Q.
So AB and BA are similar.
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