Cholesky decompositions of integral operators and the Fredholm determinant

TL;DR

This paper derives a formula for computing the Fredholm determinant of positive definite integral operators on L^2[0,1], extending the concept of Cholesky decomposition from matrices to operators.

Contribution

It introduces an equivalent Cholesky-based formula for Fredholm determinants, bridging matrix decompositions and integral operator theory.

Findings

Derived a Cholesky-based formula for Fredholm determinants

Extended matrix decomposition techniques to integral operators

Provided proofs and theoretical validation of the formula

Abstract

The Cholesky decomposition is a popular way of decomposing positive definite matrices; in particular it leads to a simple formula for computing the determinant. We present and proof an equivalent formula for computing the Fredholm determinant of a positive definite integral operator on $L^2 [0, 1]$.