Theory of Computing Systems

, 46:9

# On the Complexity of Matrix Rank and Rigidity

• Meena Mahajan
• Jayalal M. N. Sarma
Article

## Abstract

We revisit a well studied linear algebraic problem, computing the rank and determinant 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 $$\textsf{L}$$ . We show that computing the permanent and determinant of tridiagonal matrices over ℤ is in $$\textsf {Gap} \textsf {NC}^{1}$$ and is hard for $$\textsf {NC}^{1}$$ . 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 $$\textsf {NP}$$ -hard over ℚ and that a certain restricted version is $$\textsf {NP}$$ -complete. Restricting the problem further, we obtain variations which can be computed in $$\textsf {PL}$$ and are hard for $$\textsf {C}_{=}\textsf {L}$$ .

## Keywords

Complexity classes Matrix rank Determinant Matrix rigidity

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