Factorization of a prime matrix in even blocks

TL;DR

This paper explores conditions for decomposing matrices with prime-sized rows and columns into tensor products of prime matrices, linking matrix factorization to prime number conjectures and analyzing algorithm complexity.

Contribution

It introduces new necessary and sufficient conditions for matrix decomposition into prime matrices, connecting matrix theory with prime number properties.

Findings

Decomposition holds if diagonal blocks are pairwise commutative.

Algorithm complexity is $O(n^{5/2})$.

Matrix decomposition is equivalent to Goldbach's conjecture.

Abstract

In this paper, a matrix is said to be prime if the row and column of this matrix are both prime numbers. We establish various necessary and sufficient conditions for developing matrices into the sum of tensor products of prime matrices. For example, if the diagonal of a matrix blocked evenly are pairwise commutative, it yields such a decomposition. The computational complexity of multiplication of these algorithms is shown to be $O(n^{5/2})$. In the section 5, a decomposition is proved to hold if and only if every even natural number greater than 2 is the sum of two prime numbers.