Representations for the Drazin inverses of 2 × 2 block matrices☆
Introduction
For a square matrix , the Drazin inverse of A is a matrix satisfyingfor some nonnegative integer k. It is well-known that AD always exists and is unique. If Ak+1AD = Ak for some nonnegative integer k, then so does for all nonnegative integer l ⩾ k, and the smallest nonnegative integer such that the equation holds is equal to ind(A), the index of A, which is defined to be the smallest nonnegative integer such that rank (Ak+1) = rank (Ak). We adopt the convention that A0 = In, the identity matrix of order n, even if A = 0, and the index of the zero matrix is defined to be 1. We write Aπ = I − AAD. For more details we refer the reader to [1], [4].
The Drazin inverse is first studied by Drazin [23] in associative rings and semigroups. The Drazin inverse of complex matrices and its applications are very important in various applied mathematical fields like singular differential equations, singular difference equations, Markov chains, iterative methods and so on [4], [5], [24], [26], [31], [32], [35], [37].
The study on representations for the Drazin inverse of block matrices essentially originated from finding the general expressions for the solutions to singular systems of differential equations [3], [4], [5], and then stimulated by a problem formulated by Campbell [5]: establish an explicit representation for the Drazin inverse of 2 × 2 block matrices in terms of the blocks of the partition, where the blocks A and D are assumed to be square matrices. For a deeper discussion of applications of the Drazin inverse of a 2 × 2 block matrix, we refer the reader to [4], [37]. Until now, there has been no explicit formula for the Drazin inverse of general 2 × 2 block matrices. Meyer and Rose [31], [32], and independently Hartwig and Shoaf [24], [26], first gave the formulas for block triangular matrices, and since then many less restrictive assumptions are considered [2], [7], [9], [11], [12], [13], [14], [20], [22], [25], [24], [26], [29], [31], [32], [33], [34], [36], for example,
- (1)
BC = 0, BD = 0 and DC = 0 (see [20]);
- (2)
BC = 0, DC = 0 (or BD = 0) and D is nilpotent (see [25]);
- (3)
BC = 0 and DC = 0 (see [14]);
- (4)
BC = 0, BDC = 0 and BD2 = 0 (see [22]);
- (5)
BDπC = 0, BDDD = 0 and DDπC = 0 (see [22]).
Related topics are to find representations for the Drazin inverse and the generalized Drazin inverse of the sum of two matrices [6], [27], [28], [34] and operator matrices on Banach spaces [8], [10], [15], [16], [17], [18], [19], [21].
It is clear that the condition in (4) above implies that BDiC = 0 for any nonnegative integer i, by which our work is motivated. Though this condition looks like a restrictive one, it is really weaker than many ones in the literature, especially those in (1)–(5) and [16] which considers the generalized Drazin inverses, as shown in Example 4.1, Example 4.2, respectively.
In this paper, explicit expressions for the Drazin inverse of the 2 × 2 block matrix M are provided under the condition that BDiC = 0 for i = 0, 1, … , n − 1, where n is the order of D, from which many results are unified and many formulas can be derived, especially those in [14], [20], [22], [25].
For notational convenience, we define a sum to be 0, whenever its lower limit is bigger than its upper limit.
Section snippets
Preliminary
Lemma 2.1 For and matrices B, C of appropriate orders, if BDiC = 0 for i = 0, 1, … , n − 1, then BDkC = 0 and B(DD)kC = 0 for any nonnegative integer k. Proof Let f(λ) = λn − a1λn−1−⋯−an be the characteristic polynomial of D. By the Cayley–Hamilton theorem, f(D) = 0. Thusfrom which an induction on k yields BDkC = 0 for any nonnegative integer k. Since DD is expressible as a polynomial of D (see [1]), we have B(DD)kC = 0 for any nonnegative integer k. □
For simplicity of notation, we adopt the notation Ak(ε) = (Ak + εI
Main results
Let M denote a 2 × 2 block complex matrix satisfying the following conditions:Then for any nonnegative integer k, a calculation yieldswhereThenwhere i, j and l are all nonnegative integers. It is easy to verify that for any nonnegative integer i Lemma 3.1 Let , which satisfies the
Examples
The following example gives a 2 × 2 block matrix M which does not satisfy the conditions given in (1)–(5) listed in Section 1 but satisfies that in Theorem 3.5. Example 4.1 Let , where ,Since DC ≠ 0, BD2 ≠ 0 and BDD ≠ 0, the matrix M does not satisfy the conditions (1)–(5) given in Section 1. On the other hand, we can check that BDiC = 0 for i = 0, 1, 2, 3. By Theorem 3.5, we obtain
The following example shows that the condition in
Acknowledgements
The authors thank the referees for their helpful comments and suggestions.
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This work was supported by “211 Project” of Jilin University.