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).