Matrix Factorizations via the Inverse Function Theorem

TL;DR

This paper provides analytic proofs for key matrix factorizations using the inverse function theorem, establishing their dependence properties beyond traditional Gaussian elimination methods.

Contribution

It introduces a novel approach to matrix factorizations through the inverse function theorem, offering new insights into their analytic dependence.

Findings

Proofs of QR, Cholesky, and LDU factorizations via inverse function theorem

Demonstrates analytic dependence of matrix factorizations

Extends understanding beyond Gaussian elimination methods

Abstract

We give proofs of QR factorization, Cholesky's factorization, and LDU factorization using the inverse function theorem. As a consequence, we obtain analytic dependence of these matrix factorizations which does not follow immediately using Gaussian elimination.