Dynamic matrix rank

Abstract

We consider maintaining information about the rank of a matrix under changes of the entries. For n × n matrices, we show an upper bound of O(n 1.575) arithmetic operations and a lower bound of Ω(n) arithmetic operations per change. The upper bound is valid when changing up to O(n 0.575) entries in a single column of the matrix. Both bounds appear to be the first non-trivial bounds for the problem. The upper bound is valid for arbitrary fields, whereas the lower bound is valid for algebraically closed fields. The upper bound uses fast rectangular matrix multiplication, and the lower bound involves further development of an earlier technique for proving lower bounds for dynamic computation of rational functions. © Springer-Verlag Berlin Heidelberg 2006.