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

Novel Matrix Hit and Run for Sampling Polytopes and Its GPU Implementation