# Factorial of a matrix: what could be the use of it?

Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma function.

What use might it be to take the factorial of a matrix? Do any applications come to mind, or does this – for now* – seem to be restricted to the domain of recreational mathematics?

(*Until e.g. theoretical physics turns out to have a use for this, as happened with Calabi–Yau manifolds and superstring theory…)

I could find the following references that take the use of matrix factorial for a concrete applied context:

Coherent Transform, Quantization and Poisson Geometry: In his book Mikhail Vladimirovich Karasev uses for example the matrix factorial for hypersurfaces and twisted hypergeometric functions.

Artificial Intelligence Algorithms and Applications: In this conference proceeding the Bayesian probability matrix factorial is used in the context of classification of imputation methods for missing traffic data.

Fluid model: Mao, Wang and Tan deal in their paper (Journal of Applied Mathematics and Computing) with a fluid model driven by an $$M/M/1$$ queue with multiple exponential vacations and $$N$$-policy.

Construction of coherent states for multi-level quantum systems: In their paper “Vector coherent states with matrix moment problems” Thirulogasanthar and Hohouéto use matrix factorial in context of quantum physics.

Algorithm Optimization: Althought this is a more theoretic field of application, I would like to mention this matrix factorial use case as well. Vladica Andrejić, Alin Bostan and Milos Tatarevic present in their paper improved algorithms for computing the left factorial residues $$!p=0!+1!+\ldots+(p−1)!\bmod{p}$$. They confirm that there are no socialist primes $$p$$ with $$5. You may take a look into an arXiv version of this paper.