The product form for the inverse in the simplex method

Abstract

Abstract : When a matrix is represented as a product of 'elementary' matrices, the matrix, its transpose, its inverse and inverse transpose are readily available for vector multiplication. By an 'elementary matrix' is meant one formed from the identity matrix by replacing one column; thus an elementary matrix can be compactly recorded by the subscript of the altered column and the values of the elements in it. In the revised simplex method both the inverse and inverse transpose of a 'basic' matrix are needed; more significant, however, is the fact that each interation replaces one of the columns of the basis. In the product form of representation, this change can be conveniently effected by multiplying the previous matrix by an elementary matrix; thus, only one additional column of information need be recorded with each iteration. This approach places relatively greater emphasis on 'reading' operations than 'writing' and thereby reduces computation time. Using the I.B.M. Card Programmed Calculator, a novel feature results: when the inverse matrix is needed at one stage and its transpose at another, this is achieved simply by turning over the deck of cards representing the inverse.