Integer factorization of a positive-definite matrix

TL;DR

This paper proves that any positive-definite matrix can be decomposed into a sum of outer products of integer vectors with bounded entries, linking matrix properties to integer vector representations.

Contribution

Introduces a novel decomposition of positive-definite matrices into integer vector outer products with bounded entries, connecting matrix condition number and dimension.

Findings

Any positive-definite matrix can be expressed as a positive linear combination of integer outer products.

The entries of these integer vectors are bounded by the geometric mean of the matrix's condition number and its dimension.

This decomposition provides a new perspective on matrix structure and integer representations.

Abstract

This paper establishes that every positive-definite matrix can be written as a positive linear combination of outer products of integer-valued vectors whose entries are bounded by the geometric mean of the condition number and the dimension of the matrix.