Summary: The complexity of matrix rank and feasible systems of linear
Eric Allendery Robert Bealsz Mitsunori Ogiharax
We characterize the complexity of some natural and important problems in linear
algebra. In particular, we identify natural complexity classes for which the problems of
a determining if a system of linear equations is feasible and b computing the rank of
an integer matrix, as well as other problems, are complete under logspace reductions.
As an importantpart of presenting this classi cation, we show that the exact count-
ing logspace hierarchy" collapses to near the bottom level. We review the de nition of
this hierarchy below. We further show that this class is closed under NC1
andthat it consists ofexactly those languagesthat have logspace uniformspan programs
introduced by Karchmer and Wigderson over the rationals.
In addition, we contrast the complexity of these problems with the complexity of
determining if a system of linear equations has an integer solution.
The motivation for this work comes from two quite di erent sources. The rst and most
obvious source is the desire to understand the complexity of problems in linear algebra;
our results succeed in meeting this goal. The other, less obvious, source is the desire