The Turing factorization of a rectangular matrix

Abstract

The Turing factorization is a generalization of the standard LU factoring of a square matrix. Among other advantages, it allows us to meet demands that arise in a symbolic context. For a rectangular matrix A, the generalized factors are written PA = LDU R, where R is the row-echelon form of A. For matrices with symbolic entries, the LDU R factoring is superior to the standard reduction to row-echelon form, because special case information can be recorded in a natural way. Special interest attaches to the continuity properties of the factors, and it is shown that conditions for discontinuous behaviour can be given using the factor D. We show that this is important, for example, in computing the Moore-Penrose inverse of a matrix containing symbolic entries.We also give a separate generalization of LU factoring to fraction-free Gaussian elimination.