The Explicit Formula for the Moore-Penrose Inverse of a 2×2 Block Matrix

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(School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China)

Abstract: The representation for the Moore-Penrose inverse of the matrix

Keywords: Block matrix; Moore-Penrose inverse; Linear equation

§1.Introduction

Throughout this paper, Cm×ndenotes the set of allm×ncomplex matrices andIndenotes the identity matrix of sizen.ForA∈Cm×n, the Moore-Penrose inverse ofA, denoted byA†, is the unique solutionX ∈Cn×msatisfying the following four equations:

In 1991, Miao [6] established a formula of the Moore-Penrose inverse of a 2×2 block matrixMby using the{1,3}-inverse of the matrixMM∗.Baksalary and Trenkler [2] derived the formula for the Moore-Penrose inverse of a columnwise partitioned matrix.Yuan and Dai [8]provided the expression for the Moore-Penrose inverse of a columnwise partitioned matrix by using the theory of solving linear equations.Hung and Markham [5] deduced the formula of Moore-Penrose inverse of a 2×2 block matrix by using

Deng and Du [4] acquired the Moore-Penrose inverses of a 2×2 block operator valued matrices with specified properties on a Hilbert space.Yan [7] obtained an explicit expression of the Moore-Penrose inverse in terms of the partitioned matrix by using a full rank factorization of a 2×2 block matrix.In this paper, we will use the method presented in paper [8] to derive

Compared with the approach proposed in paper [5], the treatment in this paper is more concise and intuitive.

§2.Main results

To begin with, we introduce some lemmas.

Lemma 2.1.[3]If A∈Cm×q and b∈Cm.Then the equation Ay=b has a solution y ∈Cq if and only if AA†b=b,in which case,the general solution of the equation is y=A†b+FAz,where z ∈Cq is an arbitrary vector.

Lemma 2.2.[1]Let A∈Cm×n be partitioned as A=[A1,A2],then the following statements are equivalent:

(a)R(A1)∩R(A2)={0},

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(b)R(A1∗)=R(A1∗EA2),

(c)R(A2∗)=R(A2∗EA1).

Lemma 2.3.[1]Let A∈Cm×n be partitioned as A=[A1,A2],then the following statements are equivalent:

(a)R(A1)∩R(A2)={0},

(b) (EA2A1)†=A1†-A1†A2(EA1A2)†,

(c) (EA1A2)†=A2†-A2†A1(EA2A1)†.

Known by the theory of linear equations [3]: a vectorxis a least-squares solution ofAx=bif and only ifxis a solution ofA∗Ax=A∗b.This theory will be used to derive the least-squares solutions of the linear equation

wherex∈Cn,y ∈Cp,e∈Cmandf ∈Cq.Clearly, the normal equation of (2.1) is

which can be equivalently written as

whereH=A∗A+C∗C,G=A∗B+C∗D,K=B∗B+D∗D.By Lemma 2.1, the general solution of (2.2) with unknown vectorxis

whereuis an arbitrary vector.Substituting (2.4) into (2.3) yields

By Lemma 2.1, the general solution of (2.5) is

where

andvis an arbitrary vector.Substituting (2.6) into (2.4), we can obtain

where

Thus, the general least-squares solution of (2.1) is given by

Now, we will consider the minimum-norm least-squares solution of (2.1).By (2.10), we have

It is easily known that‖x‖2+‖y‖2=min if and only if

Namely,

where Γ=Q+RFLG∗H†J, Λ=P+RFLG∗H†T.

The following two representations of

can be obtained from (2.13), (2.14) and the uniqueness of the Moore-Penrose inverse:

§3.Some special cases

If putting

Proof.IfR(A∗)⊆R(C∗), then, there is an arbitrary matrixY ∈Cq×msuch thatA∗=C∗Y.By Corollary 3.4,M=R1FCA∗(B∗)†=0.Therefore, the relation of (3.4) follows.