A New Method for A Rank Subtractivity Formula

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

§1. Introduction

Throughout this paper,we denote the real m×n matrix space by Rm×n.and R(A)stand for the transpose and the range space of a given matrix A,respectively.Inrepresents the identity matrix of size n.A matrix X is called a g-inverse of A,denoted by X=A−,if it satis fies AXA=A.The collection of all possible g-inverses of A is denoted by{A−}.

Let A be an m by n matrix of rank l,and let M and N be m by k and n by q matrices,respectively,where k is not necessarily equal to q or rank

to hold.This rank subtractivity formula along with the condition under which it holds is called the extended Wedderburn-Guttman theorem.The Wedderburn-Guttman theorem[5-7]:

is used extensively in numerical linear algebra[8-9]as a rank reduction method,and in psychometrics[10-11],and statistics[12]as a means of extracting components which are known linear combinations of observed variables.

In this note,by applying the full rank decomposition ofrank(A)=rank(F)=rank(G)=l)and the product singular value decomposition of the matrix pairsome conditions for the validity of the rank subtractivity formula(1)are obtained.Compared with the approach proposed in[2],the method in this paper is more concise and easy to perform.

§2.Main Results

To begin with,we introduce two lemmas.

Lemma 1(ThePSVDTheorem[13,14])Given matrices X ∈ Rk×land Y ∈ Rl×q.Then there exist orthogonal matricesand a nonsingular matrix W ∈ Rl×lsuch that

where

and

Lemma 2(Invariance)[15]Let,and F ∈ Rp×q.Then the product DE−F does not depend on the choice of E−∈ {E−}if and only if D=0,or F=0,orand

Let FG be a full rank decomposition of the matrix A∈Rm×n,that is,

where F ∈ Rm×l,G ∈ Rl×n,l=rank(A)=rank(F)=rank(G).Assume thatit follows from the nonsingularity ofand,we can get

By Lemma 1,we have

Assume that

where Z11=S−2.It follows from(6)that

Thus,we can get

On the other hand,

By(10),we have

From(9)and(11),we have proved the following result.

Theorem 1Assume that FG is a full rank decomposition of the matrix A ∈ Rm×n,that is,F ∈ Rm×l,G ∈ Rl×n,l=rank(A)=rank(F)=rank(G).Letand let the product singular value decomposition ofbe given by(2).If the matrices Zij,i,j=1,2,3 are given by(7),then the relation of(1)holds if and only if

Letit follows from Lemma 1 that

From(12),(13)and(14),we can get Theorem 2 Letand A=FG be a full rank decomposition of the matrix A ∈ Rm×n,then the relation of(1)holds if and only if

or equivalently,

Applying Lemma 2,we can easily get Theorem 3 MatrixA is invariant over the choice of(MAN)−if and only ifandhold.

By Lemma 1,andt1=0.If r1=0,then we have

If t1=0,then we have

From(7),(13),(14),(17)and(18),we can easily obtain

Theorem 4The relation of(1)holds irrespective of(MAN)−if and only if rank=rankor rank(GN)=rank

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