Fast matrix rank algorithms and applications

We consider the problem of computing the rank of an m × nmatrix A over a field. We present a randomized algorithm to find a set of r = rank(A) linearly independent columns in Õ(|A | + rω) 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(mnrω−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