A note on the group inverse of some 2 × 2 block matrices over skew fields☆
Introduction
It is well known that the representations for the Drazin (group) inverse of block matrices are very important not only in matrix theory, but also in singular differential and difference equations, probability statistical, numerical analysis, game theory, econometrics, control theory and so on (see [1], [2], [3], [4], [5], [6], [19], [20]). In 1979, Campbell and Meyer proposed an open problem in [20], that is to find the explicit representation for the Drazin (group) inverse of the block matrix, where A and D are square matrices. Until now, this problem has not been solved completely. However, under some conditions, there have been some results about this problem (see [7], [8], [9], [10], [11], [23], [24], [25]). In 1983, on the background of second-order systems of differential equations, Campbell in [4] proposed an open problem to give an explicit representation for the Drazin inverse of a 2 × 2 anti-triangular block matrix , where C is a square matrix and 0 is a square zero matrix. And this open problem has not been solved yet. Furthermore, the Drazin (group) inverse of this kind of block matrix has some applications in other fields. For example, it can be applied to solve constrained optimization problems (KKT Linear Systems), and also can be used in finding the solution of differential equations (see [20], [21], [22]). About this open problem, there have been some results under some certain conditions (see [10], [11], [12], [13], [14], [15], [16], [17], [18], [23], [24], [25]).
Specially, in papers [15], [16], [18], [24], the existence and the representations of the group inverse for the following block matrices are researched:
- (i)
(see [24]);
- (ii)
(see [15]);
- (iii)
(see [16]);
- (iv)
, where k and l are positive integers (see [18]);
- (v)
, where non-zero elements c1, c2 are in the center of K and A♯, B♯ exist (see [18]).
Inspired by the above results, we mainly consider a class of 2 × 2 anti-triangular block matrices with the formwhere A, Y ∈ Km×n, X, B ∈ Kn×m. Clearly, this form include all cases in the above results (i)–(v).
For a matrix A ∈ Kn×n, the matrix X ∈ Kn×n satisfyingis called the group inverse of A and is denoted by X = A♯. By [1], A♯ exists if and only if rank(A) = rank(A2). If A♯ exists, then it is unique. Sinceby the existence of group inverse, we get M♯ exists if and only ifIn this paper, we give the necessary and sufficient conditions to the existence and the representations of the group inverse for block matrix when A, B, X, Y satisfy one of the following conditions:
- (1)
A, B, X, Y ∈ Kn×n, XA = AX and X is invertible, A♯ exists;
- (2)
Y = 0, A ∈ Km×n, X, B ∈ Kn×m and rank(B) ⩾ rank(A);
- (3)
Y = 0 and replace X with XB, A ∈ Km×n, B ∈ Kn×m, X ∈ Kn×n.
We mainly show three theorems as following: Theorem 1.1 Suppose , where A, B, X, Y ∈ Kn×n, A♯ exists, XA = AX and X is invertible, P = TB, then M♯ exists iff rank(B) = rank(BP); if M♯ exists, then P♯ exists and , where
Theorem 1.2
Suppose and rank(B) ⩾ rank(A), then
- (i)
M♯ exists iff rank(A) = rank(B) = rank(AB) = rank(BA);
- (ii)
if M♯ exists, then , where
Theorem 1.3
Suppose , then
- (i)
M♯ exists iff rank(A) = rank(B) = rank(AB) = rank(BA);
- (ii)
if M♯ exists, then .
The above three theorems generalize the main results in [15], [16], [18], [24], respectively. In fact, in Theorem1.1, if B♯ exists, X = c1In, Y = c2In, and 0 ≠ c1, c2 ∈ C(K), it is just the result in Theorem 3.2 of [18]. In order to show this, we need to prove equations S♯ = BB♯P♯B♯ and BS♯B = BP♯, where S = BP.
Suppose rank(B) = r, since B♯ exists, there exist invertible matrices Q ∈ Kn×n and B1 ∈ Kr×r such thatLetthenBy direct computation, the results hold. In Theorem1.1, let X = In, Y = 0, A2 = A, we also get Theorem 3.1 in [15]; in Theorem1.2, let m = n, X = In, we have Theorem3.1 in [16]; in Theorem1.3, let m = n, X = Ak−1Bl−1, we get Theorem 3.5 in [18] and if X = 0, then we get Theorem 2 in [24]; if we replace X with XB, and apply equation BA(BA)♯B = B, the formulas of Theorem1.3(ii) is obtained, (the proof of BA(BA)♯B = B in Lemma2.2). In a word, we not only extend the main results in [15], [16], [18] and [24], but also find the unity of the formulas about Theorem 3.1 in [16] and Theorem 3.5 in [18].
In this paper, we mainly use the knowledge of range and null space, this can avoid complicated decomposition. We end this section by some notations. Let K be any skew field, Km×n and C(K) = {c : c ∈ K, ck = kc, for all k ∈ K} be the set of all m × n matrices over K and the center of K, respectively. For A ∈ Km×n, rank(A), R(A) = {Ax : x ∈ Kn} and N(A) = {x ∈ Kn : Ax = 0} are called the rank, the range and the null space of A, respectively. Aπ denotes I − AA♯.
Section snippets
Some lemmas
In order to prove the main results, we give some lemmas. Lemma 2.1 Let A ∈ Km×n, B ∈ Kn×m, (AB)♯ and (BA)♯ exist, then the following equalities hold: (AB)♯ = A((BA)♯)2B; (BA)♯ = B((AB)♯)2A; (AB)♯A = A(BA)♯; (BA)♯B = B(AB)♯; (AB)♯AB = A(BA)♯B; (BA)♯BA = B(AB)♯A.
Proof
The proof is similar to the proof of lemma in paper [12]. □
Lemma 2.2
Let A ∈ Km×n, B ∈ Kn×m, and rank(A) = rank(B) = rank(AB) = rank(BA), then (AB)♯ and (BA)♯ exist, and the following equalities hold:
- (g)
A(BA)♯BA = (AB)♯ABA = A;
- (h)
B(AB)♯AB = (BA)♯BAB = B.
Proof
From [13] (or [18]), we know that AB and BA are
The proof of Theorem1.1
Proof From (1.2), we see thatSincethen M♯ exists if and only if rank(B) = rank(BP). By rank(B) = rank(BP) and R(BP) ⊂ R(B), we have R(B) = R(BP), then R(P2) = R(TBP) = TR(BP) = TR(B) = R(TB) = R(P). This implies that rank(P) = rank(P2). So P♯ exists. Let , where
Examples
Now, we provide some examples to show that our results are generalizations of other results in papers [15], [16], [18]. Example 4.1 Let the real quaternion skew field K = {a + bi + cj + dk}, where a, b, c, d are real numbers, andIt is easy to verify that XA = AX and X is invertible. By computation, we know that A♯, B♯ exist, andSince rank(B) = rank(BP) = 1, then by Theorem 1.1, we know
Acknowledgments
The authors thank two anonymous reviewers for their helpful comments.
References (25)
- et al.
On group inverse of singular Toeplitz matrices
Linear Algebra Appl.
(2005) - et al.
On group inverses of M-matrices with uniform diagonal entries
Linear Algebra Appl.
(1999) - et al.
A note on the representations for the Drazin inverse of 2 × 2 block matrices
Linear Algebra Appl.
(2007) A note on the representation for the Drazin inverse of 2 × 2 block matrices
Linear Algebra Appl.
(2008)- et al.
A note on the Drazin inverse of an anti-triangular matrix
Linear Algebra Appl.
(2009) Generalized Drazin inverses of anti-triangular block matrices
J. Math. Anal. Appl.
(2010)- et al.
Group inverse for a class 2 × 2 block matrices over skew fields
Appl. Math. Comput.
(2008) - et al.
Group inverse for the block matrices with an invertible sub-block
Appl. Math. Comput.
(2009) - et al.
Reverse order law of group inverses of products of two matrices
Appl. Math. Comput.
(2004) Involuntary function and generalized inverses of matrices on skew fields
Northeast Math.
(1987)
Matrix and Operator Generalized Inverse
Generalized Inverses: Theory and Applications
Cited by (18)
Representations for the group inverse of anti-triangular block operator matrices
2014, Linear Algebra and Its ApplicationsCitation Excerpt :In the finite-dimensional case, a matrix is presented as follows, which does not satisfy conditions of the main results of [2,4,5,9,19,32].
Further results on the group inverses and Drazin inverses of anti-triangular block matrices
2012, Applied Mathematics and ComputationOn the group inverse for the sum of matrices
2014, Journal of the Australian Mathematical SocietyRepresentations for the Drazin inverse of an anti-triangular block operator matrix E with ind(E) ≤ 2
2018, Linear and Multilinear AlgebraThe existence and the representations for the group inverse of block matrices under some conditions
2016, Operators and Matrices