Abstract We propose a novel factorization of a non-singular matrix P , viewed as a 2 × 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. Finally, we briefly explain one application of this factorization: the design of optimal circuits for a certain class of streaming permutations.
Alternative block decompositions A common way to handle the case where @math is singular is to use a permutation matrix @math that reorders the columns of @math such that the new principal upper submatrix is non-singular @cite_9 . Decomposition then becomes: @math However, @math needs to swap columns with index @math ; thus @math does not have the required form considered in our work.
@cite_9
1542938076
Historical Introduction: Issai Schur and the Early Development of the Schur Complement.- Basic Properties of the Schur Complement.- Eigenvalue and Singular Value Inequalities of Schur Complements.- Block Matrix Techniques.- Closure Properties.- Schur Complements and Matrix Inequalities: Operator-Theoretic Approach.- Schur complements in statistics and probability.- Schur Complements and Applications in Numerical Analysis.