  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.
