On the Simultaneous Numerical Inversion of a Matrix and All its Leading Submatrices

where X and Y are square matrices whose diagonal elements (where i = j) are unity, and whose subdiagonal elements (where i > j) are zero, whilst D is a diagonal matrix, that is, one whose off-diagonal elements (where i ?¿ j) are zero [1]. Where this factorization exists, it is unique [1]. Where it does not exist, there exists another factorization of the form (1) in which the matrices X and Y differ from the foregoing specification by having their rows in some way interchanged [2]. Let the matrices appearing in relation (1) all be partitioned after the rth row and column, the resultant leading submatrices of order r by r being denoted by Ar, X/, Dr, Yr, where r = 1, 2, • • -, n — 1. Then it is easily seen that