Computing the Rank Profile Matrix

TL;DR

This paper introduces the concept of the rank profile matrix, a new invariant that captures all row and column rank profile information of a matrix and its sub-matrices, and shows how to compute it efficiently using adapted Gaussian elimination algorithms.

Contribution

It defines the rank profile matrix invariant, studies conditions for Gaussian elimination to reveal this information, and adapts classical algorithms for efficient computation.

Findings

The rank profile matrix summarizes all rank profile information.

Classical CUP decomposition can be adapted to compute the rank profile matrix.

Elementary post-processing can recover echelon forms from PLUQ decompositions.

Abstract

The row (resp. column) rank profile of a matrix describes the staircase shape of its row (resp. column) echelon form. In an ISSAC'13 paper, we proposed a recursive Gaussian elimination that can compute simultaneously the row and column rank profiles of a matrix as well as those of all of its leading sub-matrices, in the same time as state of the art Gaussian elimination algorithms. Here we first study the conditions making a Gaus-sian elimination algorithm reveal this information. Therefore, we propose the definition of a new matrix invariant, the rank profile matrix, summarizing all information on the row and column rank profiles of all the leading sub-matrices. We also explore the conditions for a Gaussian elimination algorithm to compute all or part of this invariant, through the corresponding PLUQ decomposition. As a consequence, we show that the classical iterative CUP decomposition algorithm can actually be adapted to compute the rank profile matrix. Used, in a Crout variant, as a base-case to our ISSAC'13 implementation, it delivers a significant improvement in efficiency. Second, the row (resp. column) echelon form of a matrix are usually computed via different dedicated triangular decompositions. We show here that, from some PLUQ decompositions, it is possible to recover the row and column echelon forms of a matrix and of any of its leading sub-matrices thanks to an elementary post-processing algorithm.