Decell's finite algorithm for generalized inverses and tests to determine the rank of a matrix

This paper investigates the analytical and computational properties of Decell's[1] finite algorithm for determining the Moore-Penrose generalized inverse of a rectangular matrix. In Kalaba et al.[3] we show how Decell's algorithm, given by a finite sequence of matrices and scalars to be computed recursively, can be useful in the development of the algebraic properties of the Moore-Penrose generalized inverse. In this paper a complete characterization of the above sequence is given and proved. The analytical results are then used in developing three tests to determine the rank of a rectangular matrix. These tests (based on the computed matrices and scalars of the sequence) are empirically investigated and proved to be accurate.