Cholesky decomposition of positive semidefinite matrices over commutative semirings

TL;DR

This paper proves that symmetric strongly invertible matrices with nonnegative numerical range over a commutative semiring can be decomposed using Cholesky decomposition, extending classical results to a broader algebraic setting.

Contribution

It establishes the existence of Cholesky decomposition for a class of matrices over commutative semirings, generalizing known results from fields to semirings.

Findings

Cholesky decomposition exists for symmetric strongly invertible matrices over commutative semirings.

Matrices with nonnegative numerical range admit Cholesky decomposition in this setting.

The result broadens the applicability of Cholesky factorization beyond traditional algebraic structures.

Abstract

We prove that over a commutative semiring every symmetric strongly invertible matrix with nonnegative numerical range has a Cholesky decomposition.