On the Complexity of Rank and Rigidity

Given a matrix M over a ring K, a target rank r and a bound k, we want to decide whether the rank of M can be brought down to below r by changing at most k entries of M . This is a decision version of the well-studied notion of matrix rigidity. We examine the complexity of the following related problems and obtain completeness results for small (counting logspace or smaller) classes: (a) computing the determinant, testing singularity, and computing the rank for matrices with special structure, (b) determining whether k ∈ O(1) changes to a matrix suffice to bring its rank below a specified value, and (c) constructing a singular matrix closest (in a restricted sense) to the given matrix. We then consider bounded rigidity, where the magnitude of individual changes is bounded by a pre-specified value, and show NP hardness in general, and tighter bounds in special cases. We also extend the rigidity lower and upper bounds for the full-1s lower triangular matrices to full-1s extended lower triangular matrices, with a small gap between the two.