A Simplified Proof For The Application Of Freivalds' Technique to Verify Matrix Multiplication

TL;DR

This paper presents a simplified proof for verifying matrix multiplication using Freivalds' technique, along with generalizations and relaxed assumptions, enhancing understanding and applicability of the method.

Contribution

It offers a more straightforward proof of Freivalds' verification algorithm and extends the technique with broader generalizations and relaxed conditions.

Findings

Simplified proof of Freivalds' matrix verification

Generalizations of the verification technique

Relaxed assumptions for broader applicability

Abstract

Fingerprinting is a well known technique, which is often used in designing Monte Carlo algorithms for verifying identities involving ma- trices, integers and polynomials. The book by Motwani and Raghavan [1] shows how this technique can be applied to check the correctness of matrix multiplication -- check if AB = C where A, B and C are three nxn matrices. The result is a Monte Carlo algorithm running in time $Theta(n^2)$ with an exponentially decreasing error probability after each indepen- dent iteration. In this paper we give a simple alternate proof addressing the same problem. We also give further generalizations and relax various assumptions made in the proof.