Formalizing the Cholesky Factorization Theorem

Abstract

We present a formal proof of the Cholesky Factorization Theorem, a fundamental result in numerical linear algebra, by verifying formally a Cholesky decomposition algorithm in ACL2. Our mechanical proof of correctness is largely automatic for two main reasons: (1) we employ a derivation which involves partitioning the matrix to obtain the desired result; and (2) we provide an inductive invariant for the Cholesky decomposition algorithm. To formalize (1), we build support for reasoning about partitioned matrices. This is a departure from how typical numerical linear algebra algorithms are presented, i.e. via excessive indexing. To enable (2), we build a new recursive recognizer for a matrix to be Cholesky decomposable and mathematically prove that the recognizer is indeed necessary and sufficient. Guided by the recognizer, ACL2 automatically inducts and verifies the Cholesky decomposition algorithm. We also present our ACL2-based formalization of the decomposition algorithm itself, and discuss how to bridge the gap between verifying a decomposition algorithm and proving the Cholesky Factorization Theorem. To our knowledge, this is the first formalization of the Cholesky Factorization Theorem.