Necessary and Sufficient Conditions for the Existence of an LU Factorization for General Rank Deficient Matrices

TL;DR

This paper provides a complete characterization of when LU factorizations exist for all square matrices, including singular and rank-deficient cases, without permutations, and extends to rank-revealing and constrained factorizations.

Contribution

It establishes necessary and sufficient conditions for LU factorization existence for general matrices, simplifies proofs, and extends results to rank-revealing and constrained LU factorizations.

Findings

LU factorization exists iff nullity conditions are met for all leading principal submatrices.

Provides explicit sparsity bounds for L and U factors.

Derives conditions for factorizations with unit triangular factors.

Abstract

We establish necessary and sufficient conditions for the existence of an LU factorization $A=LU$ for an arbitrary square matrix $A$, including singular and rank-deficient cases, without the use of row or column permutations. We prove that such a factorization exists if and only if the nullity of every leading principal submatrix is bounded by the sum of the nullities of the corresponding leading column and row blocks. While building upon the work of Okunev and Johnson, we present simpler, constructive proofs. Furthermore, we extend these results to characterize rank-revealing factorizations, providing explicit sparsity bounds for the factors $L$ and $U$. Finally, we derive analogous necessary and sufficient conditions for the existence of factorizations constrained to have unit lower or unit upper triangular factors.