Fast matrix rank algorithms and applications

Abstract

We consider the problem of computing the rank of an m × n matrix A over a field. We present a randomized algorithm to find a set of r = rank( A ) linearly independent columns in Õ (| A | + r ω ) field operations, where | A | denotes the number of nonzero entries in A and ω < 2.38 is the matrix multiplication exponent. Previously the best known algorithm to find a set of r linearly independent columns is by Gaussian elimination, with deterministic running time O ( mnr ω-2 ). Our algorithm is faster when r < max{ m , n }, for instance when the matrix is rectangular. We also consider the problem of computing the rank of a matrix dynamically, supporting the operations of rank one updates and additions and deletions of rows and columns. We present an algorithm that updates the rank in Õ ( mn ) field operations. We show that these algorithms can be used to obtain faster algorithms for various problems in exact linear algebra, combinatorial optimization and dynamic data structure.