  We propose and analyze a new Markov Chain Monte Carlo algorithm that
generates a uniform sample over full and non-full dimensional polytopes. This
algorithm, termed "Matrix Hit and Run" (MHAR), is a modification of the Hit and
Run framework. For the regime $n^{1+\frac{1}{3}} \ll m$, MHAR has a lower
asymptotic cost per sample in terms of soft-O notation ($\SO$) than do existing
sampling algorithms after a \textit{warm start}. MHAR is designed to take
advantage of matrix multiplication routines that require less computational and
memory resources. Our tests show this implementation to be substantially faster
than the \textit{hitandrun} R package, especially for higher dimensions.
Finally, we provide a python library based on Pytorch and a Colab notebook with
the implementation ready for deployment in architectures with GPU or just CPU.
