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

TL;DR

This work establishes a complete theory for when a square matrix $A$ admits an LU factorization $A=LU$ without row/column permutations, including singular and rank-deficient cases. It introduces and proves a nullity-based condition: for every leading block size $k$, $\mathrm{null}(A[1:k,1:k]) \le \mathrm{null}(A[:,1:k]) + \mathrm{null}((A[1:k,:])^T)$, which is shown to be necessary and sufficient. The authors provide constructive, inductive proofs and extend the framework to rank-revealing factorizations with explicit sparsity bounds for $L$ and $U$, as well as analogous results for unit lower or unit upper triangular factors. The results preserve the original index structure, enabling causal-interpretation and efficient storage in applications where pivoting would disrupt structure, and they include practical Python implementations for both general and unit-triangular cases.

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.

Theorems & Definitions (27)

• Theorem 1.1: Partial Pivoting
• proof
• Corollary 1.2
• proof
• Definition 1.3
• Theorem 1.4: Full Pivoting
• proof
• Proposition 1.5
• proof
• Theorem 2.2