# Generalized Inverses of Matrix Products: From Fundamental Subspaces to Randomized Decompositions

arXiv: https://arxiv.org/abs/2602.00386 · HTML: https://mlsanity.com/paper/2602.00386/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 36 references that are in the map.
- First posted on arXiv 30 Jan 2026.

## Builds on (most cited first, 5 of 5)

- [Data-Driven Sparse Sensor Placement for Reconstruction](https://mlsanity.com/paper/1701.07569/) (2017)
- [Randomized Numerical Linear Algebra : A Perspective on the Field With an Eye to Software](https://mlsanity.com/paper/2302.11474/) (2023)
- [A New Notion of Effective Resistance for Directed Graphs-Part II: Computing Resistances](https://mlsanity.com/paper/1310.5168/) (2013)
- [Optimal Sensor and Actuator Selection using Balanced Model Reduction](https://mlsanity.com/paper/1812.01574/) (2018)
- [Towards Optimal Effective Resistance Estimation](https://mlsanity.com/paper/2306.14820/) (2023)

## Abstract (arXiv)

We investigate the Moore-Penrose pseudoinverse and generalized inverse of a matrix product A=CR to establish a unifying framework for generalized and randomized matrix inverses. This analysis is rooted in first principles, focusing on the geometry of the four fundamental subspaces. We examine: (1) the reverse order law, A^+ = R^+C^+, which holds when C has independent columns and R has independent rows, (2) the universally correct formula, A^+ = (C^+CR)^+(CRR^+)^+, providing a geometric interpretation of the mappings between the involved subspaces, (3) a new generalized randomized formula, A^+p = (PTA)^+PTAQ(AQ)^+, which gives A^+p = A^+ if and only if the sketching matrices P and Q preserve the rank of A, i.e., rank(PTA) = rank(AQ) = rank(A). The framework is extended to generalized 1,2-inverses and specialized forms, revealing the underlying structure of established randomized linear algebra algorithms, including randomized SVD, the Nystr"om approximation, and CUR decomposition. We demonstrate applications in sparse sensor placement and effective resistance estimation. For the latter, we provide a rigorous quantitative analysis of an approximation scheme, establishing that it always underestimates the true resistance and deriving a worst-case spectral bound on the error of resistance differences.