Three matrix factorizations from the steps of elimination

Abstract

Every real [Formula: see text] matrix [Formula: see text] can be factored in three ways that arise from the steps of elimination: a lower triangular/upper triangular factorization [Formula: see text], a column-row factorization [Formula: see text], and a triple factorization [Formula: see text]. The column-row factorization provides both a constructive proof that the row rank [Formula: see text] of [Formula: see text] equals the column rank, and a formula for the pseudoinverse [Formula: see text] not based on the singular value decomposition. In the triple factorization, [Formula: see text] and [Formula: see text] contain the first [Formula: see text] independent columns of [Formula: see text] and the first [Formula: see text] independent rows of [Formula: see text]; [Formula: see text] is the invertible submatrix of [Formula: see text] where [Formula: see text] meets [Formula: see text]. An alternative to the traditional elimination method, using slide steps in place of the usual swap steps, identifies the first [Formula: see text] independent rows of [Formula: see text].