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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.) $\endgroup$ – Akiva Weinberger Feb 5 '16 at 1:50

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