Matrix rigidity Original Research Article

The rigidity of a matrix M is the function RM(r), which, for a given r, gives the minimum number of entries of M which one has to change in order to reduce its rank to at most r. This notion has been introduced by Valiant in 1977 in connection with the complexity of computing linear forms. Despite more than 20 years of research, very little is known about the rigidity of matrices. Nonlinear lower bounds on matrix rigidity would lead to new lower bound techniques for the computation of linear forms, e.g., for the computation of the DFT, as well as to more general advances in complexity theory. We put forward a number of linear algebra research issues arising in the above outlined context.