Compact Schemes for $A^+B$, $A^+AB$ and $AA^+B$

TL;DR

This paper introduces efficient, storage-minimizing methods for computing matrix expressions involving Moore-Penrose pseudoinverses, specifically for $A^+B$, $A^+AB$, and $AA^+B$, using only original matrix storage and simple indexing.

Contribution

It provides explicit, compact schemes for calculating these matrix expressions without additional memory allocation, enhancing computational efficiency.

Findings

Methods require only original matrix storage and simple indexing.

Explicit formulas for $A^+B$, $A^+AB$, and $AA^+B$ are derived.

Appropriate for matrices of any nonzero size.

Abstract

Explicit details are presented for calculation of $A^+B$, $A^+AB$ and $AA^+B$ where $A_{m\times n}$ is any nonzero matrix, $A^+$ is the Moore-Penrose pseudoinverse of $A$ and $B$ is any matrix of appropriate dimensions, where the quantities in question are found using only the storage originally allocated to the matrices $A$ and $B$ (together with some simple one dimensional indexing arrays).