Factorization of a prime matrix in even blocks

TL;DR

This work introduces a framework to decompose $m\times n$ matrices into sums of tensor products of prime matrices, leveraging block-structure decompositions when $n$ is composite (notably $n=ab$) and a canonical diagonal form defined via $\otimes_{a,a}$ and $\oplus_{a,a}$. It establishes conditions under which such a decomposition exists—centered on commutativity of diagonal blocks and diagonalizability—and analyzes the computational complexity, proving an $O(n^{5/2})$ bound that improves over the naive $O(n^{3})$ for standard matrix multiplication in the blocked setting. The paper further introduces semi-prime and quasi-prime matrices, showing that even-sized dimensions allow decompositions into orthogonal quasi-primes, and connects these constructions to spectral properties and conjectures about sums of primes. It extends the discussion to non-square cases and provides concrete $4\times4$ examples and a practical factorization algorithm for realizing the proposed decompositions. Overall, it offers a structured approach to represent large matrices via sums of tensor products with potential applications in statistics, physics, and wireless communications, while linking to deep number-theoretic ideas about prime partitions. All results are expressed with explicit tensor-analytic constructions and complexity analyses, providing both theoretical insight and actionable algorithms.

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.

Figures (6)

• Figure 1: The canonical diagonal form with $A = \oplus_{a,a} \left(D_{1} \otimes_{a,a} C \right)$
• Figure 2: The canonical diagonal form with $A = \oplus_{a,a} \left(D_{1\frac{1}{2}} \otimes_{a,a} C\right)$
• Figure 3: The canonical diagonal form with $A = D_{2} \cdot (B\otimes C)$.
• Figure 4: The canonical diagonal form with $A = D_{3} \cdot (B\otimes C)$.
• Figure 5: The canonical diagonal form $D_{1}^{\prime}$

Theorems & Definitions (17)

• Definition 2.1
• Theorem 3.1
• Theorem 4.1
• Definition 5.1: Semi-prime matrix
• Definition 5.2: Quasi-prime matrix
• Conjecture 5.3
• Lemma 5.4
• Corollary 5.5
• Corollary 5.6
• Theorem 5.7