Formulas for the drazin inverse of special block matrices

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Abstract

Properties of the Drazin inverse of the matrix F=IIE0, with E square, are investigated. Based on this approach, it is obtained an explicit formula for the Drazin inverse of matrices of the form A=IPtQUVt, where U, V, P and Q are n × k. The representation for AD is given in terms of k × k matrices involving the individual blocks under some conditions. Some special cases are also analyzed.

Introduction

Let us recall that the Drazin inverse of ACn×n is the unique matrix ADCn×n satisfying the relationsADAAD=AD,AAD=ADA,Al+1AD=Alforalllr,where 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 R(A) and N(A) 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 R(Aπ)=N(Ar) and N(Aπ)=R(Ar). It is well-known that Aπ = I  AAD.

In this paper, we present a formula for computing the Drazin inverse of matrices of the formA=IPtQUVt,where 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 M=IBCD, 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 matrixF=IIE0,Esquare,ind(E)=r,andIidentity.Therein, it was showed thatFD=Y1EπED+Y2EπEDE+Y2EEπ-ED+(Y1-Y2)Eπ,whereY1=j=0r-1(-1)jC(2j,j)EjandY2=j=0r-1(-1)jC(2j+1,j)Ej,and 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 nk. 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(n,n-k)=C(n,k)=nkC(n-1,k-1),k0; 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. Thenj=0knn+2jC(n+2j,j)C(2(k-j),k-j)=C(n+2k,k),integerk0.

We remark that if n = 0, the factor 00+2jC(0+2

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 n2. We introduce the notationYπ(n)=j=0r-1(-1)jC(n+2j,j)EjEπ,n-1.The 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 asFD=G+H,G=0EDEDE-ED,H=Yπ(0)Yπ(1)Yπ(1)EYπ(2)E.Notice thatYπ(n)=j=0r-1(-1)jC(n+2j,j)(EEπ)jEπ.Since 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 A=IPtQUVt, where U, V, P and Q are n × k matrices. The matrix A can be written as A = BCt, where B=0IIUQ0 and C=0I0V0P. Hence, we have [4]AD=(BCt)D=B((CtB)D)2Ct.We haveCtB=0VtI00Pt0IIUQ0=VtUVtQ00IIPtUPtQ0.

Let us extract the block matrix F=IIPtQ0 and let r = ind(PtQ). In the next theorem we will refer to this matrix F and to the following sequences:Xd(n)=i=0s(n)C(n-i,i)((PtQ)D)n-i,n0;Xd(-1)=0,Yπ(n)=j=0r-1(-1)jC(n+2j,j)(PtQ)j(PtQ)π,n-1.

Let Fij2 and (FD)ij2

Acknowledgment

This research was partly supported by the “Agencia Española de Cooperación Internacional (AECI)”, Project 201/03/P.

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