A simple proof of the product theorem for EP matrices

Linear Algebra and its Applications 294 (1999) 213±215 www.elsevier.com/locate/laa

A simple proof of the product theorem for EP matrices J.J. Koliha

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Department of Mathematics and Statistics, University of Melbourne, Parkville 3052, Australia Received 5 August 1998; accepted 16 March 1999 Submitted by B. Cain

Abstract We give a simple proof of a recent theorem of Hartwig and Katz on the product of EP matrices. Ó 1999 Elsevier Science Inc. All rights reserved. Keywords: EP matrix; Moore±Penrose inverse; Eigenprojection

The purpose of this note is to give a simple proof of a theorem recently obtained by Hartwig and Katz [3, Theorem 1] on the product of EP matrices that solved a problem open for over 25 years (see [1]). All matrices are complex and n  n. A matrix A is EP if Ay A ˆ AAy , where Ay is the Moore±Penrose inverse of A (see [2]). We need the following result. Proposition 1 [2, Theorem 4.3.1]. For any matrix A the following are equivalent: (i) A is EP. (ii) N …A† ˆ N …A †, where N …A† is the nullspace of A. (iii) R…A† ˆ R…A †, where R…A† is the range of A. (iv) Cn ˆ N …A† ? R…A†. Clearly, A is EP if and only if A is EP. We observe that every EP matrix A satis®es

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0024-3795/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5 ( 9 9 ) 0 0 0 6 6 - X

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J.J. Koliha / Linear Algebra and its Applications 294 (1999) 213±215

A ˆ UA

for some matrix U : 

…1†



(For instance, A ˆ A2 Ay ˆ A…Ay † A .) We write Ap for the eigenprojection of A at 0; if A is EP, Ap is the orthogonal projection onto N …A†. Theorem 2. Let A, B be EP matrices. Then the following are equivalent: (i) AB is EP. (ii) …AB†Ap ˆ 0 and Bp …AB† ˆ 0. (iii) N …A†  N …AB† and R…AB†  R…B†. (iv) N …AB† ˆ N …A† ‡ N …B† and R…AB† ˆ R…A† \ R…B†. 

Proof. (i) ) (ii). By (1), AB ˆ U …AB† for some matrix U. From Proposition 1 (iv) we deduce that Ap is Hermitian and that Ap A ˆ AAp ˆ 0. Then 



…AB†Ap ˆ U …AB† Ap ˆ UB A Ap ˆ UB …Ap A† ˆ 0: 

Since …AB† ˆ B A is also EP, the preceding argument applies to give  Bp …AB† ˆ ……B A †Bp † ˆ 0. (ii) ) (iii). From …AB†Ap ˆ 0 and R…Ap † ˆ N …A† follows N …A†  N …AB†, and from Bp …AB† ˆ 0 and N …Bp † ˆ R…B† follows R…AB†  R…B†. (iii) ) (iv). Suppose that N …A†  N …AB†. The map h de®ned on the quotient space N …AB†=N …B† by h…x ‡ N …B†† ˆ Bx is a vector space isomorphism of N …AB†=N …B† onto N …A† \ R…B†. Hence dim…N …A† \ R…B†† ˆ dim N …AB† ÿ dim N …B†:

…2†

From Cn ˆ N …B†  R…B† we deduce that W ˆ …N …A† \ R…B††  N …B†  N …A† ‡ N …B†  N …AB†: By (2), dim W ˆ dim N …AB†, and W ˆ N …A† ‡ N …B† ˆ N …AB†.  Next assume that R…AB†  R…B†. This implies N …B †  N ……AB† †, and by the    ®rst part of the proof, N ……AB† † ˆ N …A † ‡ N …B †. This is equivalent to R…AB† ˆ R…A† \ R…B†.  (iv) ) (i). Since N …AB† ˆ N …A† ‡ N …B† implies R……AB† † ˆ R…A † \ R…B †, we have 

R……AB† † ˆ R…A † \ R…B † ˆ R…A† \ R…B† ˆ R…AB†; and AB is EP by Proposition 1(iii).  The main result of Hartwig and Katz [3] is formulated using the row space RS…A† of a matrix A. Their result follows from Theorem 2 when we observe that RS…AB†  RS…A† () N …A†  N …AB†; RS…AB† ˆ RS…A† \ RS…B† () N …AB† ˆ N …A† ‡ N …B†:

J.J. Koliha / Linear Algebra and its Applications 294 (1999) 213±215

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Corollary 3 [3, Theorem 1]. Let A, B be EP matrices. Then the following are equivalent. (i) R…AB† ˆ R…A† \ R…B† and RS…AB† ˆ RS…A† \ RS…B†. (ii) R…AB†  R…B† and RS…AB†  RS…A†. (iii) AB is EP. References [1] T.S. Baskett, I.J. Katz, Theorems on products of EPr matrices, Linear Algebra Appl. 2 (1969) 87±103. [2] S.L. Campbell, C.D. Meyer, Generalized Inverses of Linear Transformations, Pitman, London, 1979. [3] R.E. Hartwig, I.J. Katz, On products of EP matrices, Linear Algebra Appl. 252 (1997) 339±345.