Generalized Wedderburn Rank Reduction

TL;DR

This paper extends the Wedderburn rank reduction formula by using the Moore--Penrose pseudoinverse, enabling broader applications in matrix decompositions and low-rank approximations without requiring non-singularity.

Contribution

It generalizes the Wedderburn formula with pseudoinverses, unifies various matrix decompositions, and characterizes properties of projections derived from this method.

Findings

Generalized Wedderburn reduction using Moore--Penrose pseudoinverse.

Explicit description of reductions for optimal low-rank approximation.

Range and nullspace characterization of specific matrix projections.

Abstract

We generalize the Wedderburn rank reduction formula by replacing the inverse with the Moore--Penrose pseudoinverse. In particular, this allows one to remove the non--singularity of a certain matrix from assumptions. The results implies in a straightforward way Nystroem, CUR decompositions, meta-factorization, and a result of Ameli, Shadden. We investigate which properties of the matrix are inherited by the generalized Wedderburn reduction. Reductions leading to the best low-rank approximation are explicitly described in terms of singular vectors. We give a self--contained calculation of the range and the nullspace of the projection $A(BA)^+B$ and prove that any projection can be expressed in this way.