Integral matrices as diagonal quadratic forms

TL;DR

This paper studies when diagonal quadratic forms with integer coefficients can represent all integral matrices of a given size, providing necessary and sufficient conditions for 2x2 matrices and sufficient conditions for larger sizes.

Contribution

It establishes a complete characterization for 2x2 matrices and offers new sufficient conditions for higher dimensions with coprime coefficients.

Findings

Necessary and sufficient condition for 2x2 matrices

Sufficient conditions for n×n matrices with coprime coefficients

Extension of representation criteria to higher dimensions

Abstract

In this paper, we investigate the conditions under which a diagonal quadratic form $\sum_{i=1}^{m}a_i X_i^2$ represents every $n \times n$ integral matrix, where $a_i$ ($1 \leq i \leq m$) are integers. For $n=2$, we give a necessary and sufficient condition. Also we give some sufficient conditions for each $n \geq 2$ where $a_i$ ($1 \leq i \leq m$) are pairwisely coprime.