Can one factor matrices?

Question

I know that one can factor integers as a product of prime numbers. Is there an analog of it to matrices? Can we define prime matrices such that every matrix is a product of prime matrices? Is there any applications of factorization of matrices?

Answer 1

A matrix is prime if and only if it is not a product of two matrices except for the matrix itself. You can definitely factor them and we can define a product of primes for every matrix. Although this product is not Commutative.