Hand Computation of Generalized Inverses

Abstract

A growing interest in the theory of generalized inverses ([2], [13]) and in applications of generalized inverses in such fields as engineering ([4], [5], [9], [17]) and statistics ([1], [6], [7], [12], [15]) has stimulated an interest in teaching about generalized inverses in linear algebra courses ([3], [11], [16]). When A is a nonsingular matrix, the equation Ax = b has a unique solution, namely, x = A lb. The generalized inverse A + of a singular matrix A can be motivated as a generalization of this situation. In fact, x = A + b will be the unique solution to Ax = b when a unique solution exists, will be a solution when there is more than one solution, and will be a least squares solution (i. e., will minimize IIAx b1(2) when no solution exists. Moreover, when either the solution is not unique, or the least squares solution is not unique, the generalized inverse will be the shortest solution, or the shortest least squares solution (i.e., will minimize (Ix 12, subject to the restriction that x is a solution, or that x is a least squares solution, as appropriate) [11], [16]. A geometric interpretation of the generalized inverse also helps to motivate the theory; x = A + b lies in the row space of A, while Ax = AA + b is the projection of b onto the column space of A [16]. Some instructors prefer to relegate all computations of inverses to computers. However, many instructors feel that students gain an understanding from computing an inverse by hand that cannot be gained from observation of computer output, and teach an elementary algorithm for computing inverses of matrices. This note is of interest only to the latter set of instructors. It contains an algorithm for the calculation of generalized inverses, suitable for use by students with hand-held calculators, which uses the information available when the usual (Gauss-Jordan) classroom algorithm for computing inverses has established that the inverse fails to exist. There are two equivalent procedures, which enable a student to calculate the generalized inverse using one procedure, and to check the work using the other procedure. This algorithm is meant for classroom use only; there are more accurate, computationally stable, and efficient algorithms available in various computer packages (for example, the GINV function in the MATRIX procedure of SAS [14]) for use in real world problems. A classroom discussion of the use of the results of a singular value decomposition in finding a generalized inverse is enlightening ([10], [11], [16]), but finding a singular value decomposition using only a hand-held calculator restricts attention to trivial examples when this approach is used. An attempt to invert a matrix A using the usual classroom algorithm will either yield an inverse (when A is nonsingular), or else will at least provide us with the row-echelon form of A. The columns of A corresponding to columns of the identity matrix appearing in the row-echelon form are a basis for the column space of A; we form a matrix B from these columns. As we shall s4ow later, the generalized inverse A + of A can then be expressed using B in the expression