# On Fast Matrix Inversion via Fast Matrix Multiplication

## TL;DR

This paper discusses advanced algorithms for fast matrix inversion leveraging fast matrix multiplication, analyzing their complexities, limitations, and extensions to polynomial matrices, building on foundational work by Strassen and others.

## Contribution

It provides a detailed complexity analysis of fraction free matrix inversion methods and explores their limitations, especially for multivariate polynomial matrices, extending prior foundational algorithms.

## Key findings

- Fraction free inversion methods have specific complexity bounds.
- Limitations arise when applying these methods to polynomial matrices.
- Analysis clarifies the true computational costs and constraints.

## Abstract

Volker Strassen first suggested an algorithm to multiply matrices with worst case running time less than the conventional $\mathcal{O}(n^3)$ operations in 1969. He also presented a recursive algorithm with which to invert matrices, and calculate determinants using matrix multiplication. James R. Bunch & John E. Hopcroft improved upon this in 1974 by providing modifications to the inversion algorithm in the case where principal submatrices were singular, amongst other improvements. We cover the case of multivariate polynomial matrix inversion, where it is noted that conventional methods that assume a field will experience major setbacks. Initially, the author and others published a presentation of a fraction free formulation of inversion via matrix multiplication along with motivations, however analysis of this presentation was rudimentary. We hence provide a discussion of the true complexities of this fraction free method arising from matrix multiplication, and arrive at its limitations.