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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...)

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    $\begingroup$ Perhaps it can be used to define the factorial of a quaternion? (That is, one would represent the quaternion by its matrix representation and find the factorial of that. It seems the final answer involves $\left(a\pm i\sqrt{b^2+c^2+d^2}\right)\!\!\matrix{\large!}\!$, which we know how to define.) $\endgroup$ Feb 5, 2016 at 1:50
  • $\begingroup$ This is a bit speculative, but it is probably useful in probability theory. The factorial/gamma function is a pretty common component in random variable probability densities, (think poisson, gamma, beta distributions), and generalising univariate pdfs to multivariate pdfs often involves replacing the role of a single variable with a matrix. The matrix exponential shows up a lot in probability theory, so there might be some occurrences of the matrix factorial as well. $\endgroup$ Apr 1, 2021 at 1:34

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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.

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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 $5<p<2^{40}$. You may take a look into an arXiv version of this paper.

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The factorial has a straightforward interpretation in terms of automorphisms/permutations as the size of the set of automorphisms.

One possible generalization of matrices is the double category of spans.

So an automorphism $ R ! = R \leftrightarrow R $ over a span $A \leftarrow R \rightarrow B$ ought to be a reasonable generalization.

I usually find it easier to think in terms of profunctors or relations than spans.

The residual/internal hom of profunctors $(R/R)(a, b) = \forall x, R(x, a) \leftrightarrow R(x, b) $ is a kan extension. The kan extension of a functor with itself is the codensity monad. For profunctors and spans the automorphism ought to be a groupoid (a monad in the category of endospans equipped with inverses.)

The factorial is the size of the automorphism group of a set. The automorphism group ought to generalize to a "automorphism groupoid" of a span. I suspect permutation of a matrix ought to be a automorphism groupoid enriched in Vect but this confuses me.

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