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Cholesky decompositions of integral operators and the Fredholm determinant

Niels Lundtorp Olsen

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 $L^2 [0, 1]$.

Cholesky decompositions of integral operators and the Fredholm determinant

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 .

Paper Structure

This paper contains 1 section, 2 theorems, 21 equations.

Table of Contents

  1. The Cholesky decomposition and the determinant

Key Result

Theorem 1

Let $A$ be an integral operator on $[0,1]$ with continuous kernel. Suppose $\mathbf{I}+A$ decomposes $\mathbf{I} + A = (\mathbf{I} + T) (\mathbf{I} + T^*)$ where $T$ is a triangular operator. Assume that $T$ is continuous on the closed lower triangle $\{(x,y) | 0 \leq x \leq 1, 0 \leq y \leq x \}$ (

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Lemma 1
  • proof