A Lower-Upper-Lower Block Triangular Decomposition with Minimal Off-Diagonal Ranks

TL;DR

This paper introduces a new matrix factorization that minimizes off-diagonal ranks in a 2x2 block matrix, generalizing Block LU factorization with an optimal algorithm for minimal off-diagonal ranks.

Contribution

It presents a novel factorization of non-singular matrices into three specific block-triangular matrices with minimal off-diagonal ranks, extending the Block LU factorization.

Findings

Derived lower bounds for off-diagonal ranks.

Provided an algorithm to compute optimal factorizations.

Achieved a generalization of Block LU factorization.

Abstract

We propose a novel factorization of a non-singular matrix $P$, viewed as a $2\times 2$-blocked matrix. The factorization decomposes $P$ into a product of three matrices that are lower block-unitriangular, upper block-triangular, and lower block-unitriangular, respectively. Our goal is to make this factorization "as block-diagonal as possible" by minimizing the ranks of the off-diagonal blocks. We give lower bounds on these ranks and show that they are sharp by providing an algorithm that computes an optimal solution. The proposed decomposition can be viewed as a generalization of the well-known Block LU factorization using the Schur complement.