Formulas for the drazin inverse of special block matrices
Introduction
Let us recall that the Drazin inverse of is the unique matrix satisfying the relationswhere r is the smallest non-negative integer such that rank(Ar) = rank(Ar + 1), i.e., r = ind(A), the index of A. The case when ind(A) = 1, the Drazin inverse is called the group inverse of A and is denoted by A♯.
Let and denote the range space of A and the null space of A, respectively. The eigenprojection Aπ of A corresponding to the eigenvalue 0 is the uniquely determined idempotent matrix with and . It is well-known that Aπ = I − AAD.
In this paper, we present a formula for computing the Drazin inverse of matrices of the formwhere U, V, P and Q are n × k matrices. This type of matrices are refereed in the literature as bordered matrices and they appear in linear algebra problems in some fields of applied mathematics (see [1], [5] and references therein).
The framework of considered matrices includes block matrices of the form , I the k × k identity, D the n × n matrix with rank(D) = k. In this case, D can be factorized as D = UVt, where U, V are n × k matrices with rank(U) = rank(V) = k [2]. At present time there is no known representation for the Drazin inverse of M with arbitrary blocks. Recently, expressions for MD has been given under some conditions on the blocks [8], [3].
In [3] the first and second author gave a formula for computing the Drazin inverse of the block matrixTherein, it was showed thatwhereand also a representation for (FD)2 was given. However, it would be desirable to have a representation for the general powers (FD)n. Also an expression for FπFn for arbitrary integers n will be needed to get our objective. This will be the task to develop in Section 3. First, in Section 2 we give combinatorial identities which will be used in the forthcoming section. Finally, in Section 4, we give representations of the Drazin inverse of the block matrix (1.2) in terms of the k × k matrices PtQ, VtU, PtU, VtQ, (PtQ)D and (VtU)D, under some conditions.
Section snippets
Preliminary results
Throughout this paper, for integers n and k, we denote by C(n, k) the binomial coefficient . We will make use of the following well-known identities involving binomial coefficients: C(n, k) = C(n − 1,k) + C(n − 1, k − 1); ; C(−l, k) = (−1)kC(l + k − 1, k).
In the following lemma we give a general convolution identity which we refer to [6, p. 202, formula (5.62)]. Lemma 2.1 Let n be an integer. Then
We remark that if n = 0, the factor
Properties of the Drazin inverse of the matrix F
Throughout this paper for any integer n we will denote by s(n) the integer part of . We introduce the notationThe sequence of matrices Yπ(n) allow us to write in a compact form FD and it will get involved in the powers of (FD)n.
The expression (1.4) for FD can be rewritten asNotice thatSince EEπ is a nilpotent matrix, we were motivated to work with a general nilpotent matrix in
The Drazin inverse for a class of block matrices
We consider the matrices of the form , where U, V, P and Q are n × k matrices. The matrix A can be written as A = BCt, where and . Hence, we have [4]We have
Let us extract the block matrix and let r = ind(PtQ). In the next theorem we will refer to this matrix F and to the following sequences:
Let and
Acknowledgment
This research was partly supported by the “Agencia Española de Cooperación Internacional (AECI)”, Project 201/03/P.
References (8)
- et al.
Representations of the Drazin inverse for a class of block matrices
Linear Algebra Appl.
(2005) - et al.
A Drazin inverse for rectangular matrices
Linear Algebra Appl.
(1980) - et al.
On the spectrum and pseudoinverse of a special bordered matrix
Linear Algebra Appl.
(2001) - et al.
Generalized inverses: theory and applications
(2003)