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

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.

💡 Research Summary

The paper revisits the classic Freivalds technique for verifying matrix multiplication, aiming to provide a more straightforward proof and to relax the usual uniform‑distribution assumption on the random vector used in the test. The problem setting is the standard one: given three n × n matrices A, B, and C, decide whether AB = C. A deterministic verification would require Θ(n³) time, whereas Freivalds’ Monte‑Carlo algorithm runs in Θ(n²) time with a one‑sided error that can be reduced exponentially by repetition.

Main Contributions

1. Simplified Proof of the Standard Bound – The authors restate the well‑known result that if AB ≠ C and a random binary vector r ∈ {0,1}ⁿ is chosen uniformly, then the probability that (AB)r = Cr is at most ½. Their proof proceeds by viewing matrix‑vector multiplication as a linear combination of the columns of the product matrix D = AB. They define Y = {j | dⱼ ≠ cⱼ}, note that |Y| ≥ 1, and observe that (AB)r = Cr can only happen when all components rⱼ for j ∈ Y are zero. Since each rⱼ is an independent Bernoulli(½) variable, the probability of this event is (½)^{|Y|} ≤ ½. This argument is essentially the same as the textbook proof but is presented in a more “column‑selection” language, which may be pedagogically appealing.

2. Generalization to Arbitrary i.i.d. Distributions – The second theorem claims that the uniformity requirement can be dropped: if each component of r is drawn independently from any distribution f (i.e., P