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Is it possible to generalise prime numbers to matrices? I'm trying to solve a Rubix cube in the minimum number of steps and I think this would be useful. I think it's possible to represent Rubix cube operations in the language of linear algebra or matrices. From there, maybe I can represent a solution of the Rubix cube as a product of matrices. Transforming a product of matrices into its minimum decomposition (this is where the prime version of matrices come in) should provide a 'minimum' solution.

Disclosure: this is just my intuition and I understand completely if what I just wrote doesn't make much sense).

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    $\begingroup$ Do you know about rings? $\endgroup$
    – Juan Sebastian Lozano
    Apr 26, 2017 at 2:22
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    $\begingroup$ I see, I don't have the time to write an answer right now, but generally "prime numbers" are with respect to the number system you are working in - so prime numbers make sense in the integers, but prime numbers in the rationals don't make sense, even though the integers are contained in the rationals. These number systems are called rings, and usually matrices form a ring, called a matrix ring. So what you want are the prime elements of a matrix ring. $\endgroup$
    – Juan Sebastian Lozano
    Apr 26, 2017 at 2:29
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    $\begingroup$ However, I should note that prime elements really only make sense in the case that your ring has commutative multiplication. Most matrix rings don't have commutative multiplication, though. $\endgroup$
    – Juan Sebastian Lozano
    Apr 26, 2017 at 2:31
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    $\begingroup$ I suspect that the abstract mathematics you are trying to invent is an application of group theory, not ring theory. Search group theory rubik cube and you'll find several links. Here's one: math.harvard.edu/~jjchen/docs/… $\endgroup$
    – Ethan Bolker
    Apr 26, 2017 at 2:34
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    $\begingroup$ @Christian on a more fundamental note: finding a scheme to represent Rubik manipulations with matrices is called finding a representation of the Rubik's cube group $\endgroup$
    – Ben Grossmann
    Apr 26, 2017 at 2:34

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One peculiar connection between primes and matrix products is through the definition of a so called dynamical zeta function . I am not sure this is what you are looking for, but it may give you some ideas. Here is an explicit application to products of matrices https://arxiv.org/abs/chao-dyn/9301001

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