Note on a Least Squares Inverse for a Matrix

For an m X n complex matrix A, a least squares inverse for A is used to characterize the set of all least squares solutions of an inconsistent system of equations AX = b, and two representations for a least squares inverse of a partitioned matrix are obtained. Let A be an m X n complex matrix, b an m X 1 complex vector, and S(A, b) the set of all least squares solutions for the system Ax = b. By a least squares inverse (/-inverse) for A is meant any n X m matrix A t such that for each b in E ~, A*b is in S(A, b). It is generally known that a matrix X is an /-inverse for A if and only if X satisfies AXA = A and (AX)* = AX, Penrose's first and third equations. (See [3, 5].) The purpose here is to characterize the set S(A, b) in terms of /-inverses and to present two representations of an/-inverse of a partitioned matrix. Although in general a matrix has many /-inverses, there are cases when the l-inverse is unique (and hence is A +, the generalized inverse). THEOREM 1. If A is m X n of rank r then A t is unique if and only if r = n < m. which has a unique solution if and only if r = n, and with r = n we cannot have m<n. Inhis paper [3] Penrose proves that if the system Ax = b is consistent, the general solution is given by x = A-b + (I-A-A)h, h C E n (2) where A-is any l-inverse for A (i.e. AA-A = A). However, if the system Ax = b is not consistent then eq. (2) does not suffice to produce least squares solutions to the inconsistent system. In this case an/-inverse can be utilized as follows. THEOREM 2. Let A t be any l-inverse and A-be any 1-inverse for A. Then x is in S(A, b) if and only if x can be written as x = A~b + (I-A-A)h, h E E n.