On the Complexity of Matrix Rank and Rigidity Meena Mahajan and Jayalal Sarma M.N. The Institute of Mathematical Sciences, Chennai 600 113, India. {meena,jayalal}@imsc.res.in Abstract We revisit a well studied linear algebraic problem, computing the rank and determi- nant of matrices, in order to obtain completeness results for small complexity classes. In particular, we prove that computing the rank of a class of diagonally dominant matrices is complete for L. We show that computing the permanent and determinant of tridiagonal matrices over Z is in GapNC 1 and is hard for NC 1 . We also initiate the study of computing the rigidity of a matrix: the number of entries that needs to be changed in order to bring the rank of a matrix below a given value. We show that some restricted versions of the problem characterize small complexity classes. We also look at a variant of rigidity where there is a bound on the amount of change allowed. Using ideas from the linear interval equations literature, we show that this problem is NP-hard over Q and that a certain restricted version is NP-complete. Restricting the problem further, we obtain variations which can be computed in PL and are hard for C = L. 1 Introduction A series of seminal papers by a variety of people including Valiant, Mulmuley, Toda, Vinay, Grigoriev, Cook, and McKenzie, set the stage for studying the complexity of computing matrix properties (in particular, determinant and rank) in terms of logspace computation and poly-size polylog-depth circuits. This area has been active for many years, and an NC upper bound is known for many related problems in linear algebra; see for instance [All04]. Some of the major results in this area are that computing the determinant of integer matrices is GapL-complete and that testing singularity of integer matrices is C = L-complete. In particular, the complexity of computing the rank of a given matrix over Q has been well studied. For general matrices, checking if the rank is at most r is C = L-complete [ABO99]. Complete problems for complexity classes are always promising, since they provide a set of possible techniques that are associated with the problem to attack various questions regarding the complexity class. Such results can be expected to flourish when the complete problem has well-developed tools associated with it. With this motivation, we look at special 1