A Unified Approach for the Recursive Determination of Generalized Inverses

Keywords--Genera l ized inverses, Recursive determination, Unified approach, Dynamic programming. 1. I N T R O D U C T I O N The recursive determination of a generalized inverse of a matr ix finds extensive applications in the fields of statistical inference [1-3], filtering theory, estimation theory [4], and system identification [5]. More recently, generalized inverses have found renewed applicability in the field of analytical dynamics [6,7]. The reason for the extensive applicability of recursive relations is tha t they provide a systematic method to generate 'updates ' whenever sequential addition of da ta or new information becomes available, and updated estimates which take into account this additional information are required. The recursive scheme for the computat ion of the Moore-Penrose (MP) inverse [8,9] of a matr ix was ingeniously obtained in a famous paper by Greville in 1960 [2]. Because of its extensive applicability, Greville's result is widely stated in almost every book tha t touches on the subject of generalized inverses of matrices. Yet, because of the complexity of his solution technique, Greville 's proof is seldom, if ever, quoted or outlined, even in specialized texts which deal solely with generalized inverses of matrices. For example, in books like [4,10-12], Greville's result is stated, but no constructive proof is provided, most likely because of its perceived complexity. In the same vein, Mitra and Bhimasankaram [13] provide several results for the recursive determinat ion of generalized inverses of matrices; they state their results as several Ansatze and prove them by directly verifying their validity using a number of specialized results related to generalized inverses of matrices. Their results are equivalent to those presented here. However, they provide no constructive proofs for their results and their proofs for the various types of gen0898-1221/98/$ see front matter © 1998 Elsevier Science Ltd. All rights reserved. PII: S0898-1221 (98)00247-8 Typeset by ~4.h~-TEX 126 F . E . UDWADIA AND R. E. KALABA eralized inverses have no underlying reasoning or thread running through them; only verifications of the various Ansatze are carried out. In this paper, we present a simple constructive approach inspired in part by Bellman's optimality principle, to the recursive determination of various generalized inverses of a matrix. The approach relies on a unified underlying theme and shows clearly why and how the differences in the recursive forms of the various generalized inverses arise. Thus, our results encompass those of Greville [2], and our method of proof, being constructive, provides deeper insights into the nature of the recursive determination of generalized inverses. For convenience, we introduce the following notation. Given a real matrix A, its MP-inverse G satisfies the following four conditions: (1) A G A = A, (2) G A G = G, (3) A G is symmetric, and (4) G A is symmetric. We shall denote a matr ix G which satisfies all four of these conditions by A {1'2'a'4}. Similarly, a matrix which satisfies only the first and fourth condition above shall be denoted as A {1,4} and shall be referred to as the {1,4}-inverse of A, etc. The most commonly used generalized inverses of a matrix are the MP-inverse (also denoted here as the {1,2,3,4}-inverse), the {1,3}-inverse, the {1,4}-inverse, and the {1}-inverse because these inverses are relevant to the solution x of the matrix equation A x = b or of the relation A x ~ b. We shall begin by defining these generalized inverses (as in [12]) in terms of the relevant linear relations which they help solve. The MP-inverse provides the minimum-length solution x = A{1'2's'a}b in the set of least-squares solutions of the possibly inconsistent equation A x ~ b for any b, the {1,3}-inverse provides a least-squares solution A{1,S}b to the possibly inconsistent equation A x ~ b for any b, the {1,4}-inverse provides a minimum-length solution A {1,4} b for any b for which the equation is consistent, and the {1}-inverse of A provides a solution A{1}b for any b for which the equation A x = b is consistent. This paper is concerned with these four commonly used generalized inverses defined above, which we shall denote, in general, by A*. The solution x is then expressed, in general, as A*b. Their generalized forms are given in [14]. Given a real m by k matrix Ak, one can partit ion it as [Ak-1 a] where Ak-1 consists of the first ( k 1) columns of the matrix Ak and a is its last column. The column vector a comprises 'new' or additional information, while the matrix Ak-1 comprises accumulated past data. The generalized inverse A~ of the updated matrix Ak is then sought in terms of the generalized inverse A~_ 1 of the matrix Ak-1 which corresponds to past accumulated data, and the vector a containing new or additional information. The MP-inverse of a matrix A is unique. The other generalized inverses ~{1,4} which we shall deal with here are not in general unique, and so, in what follows, by say " 'k-1 , we shall mean any one of the set of {1,4}-inverses of the matrix Ak-1. 2. M A I N R E S U L T Let Ak = [Ak-1 a] be an m by k matrix whose last column is a. Let the m-vector c = * T * T * ( I Ak_lA*k_l)a and let the m-vector d = ( A k _ l ) T A*k_la/(1 + a (Ak_ l ) A k _ l a ). Then, and A'k_ -A*k_lau T ] A* k = u T J ' [ A~_ 1 A*k_lav T ] A~¢ = vT j , for c # o (la)