Lower Bound of 22 for 3x3 Matrix Multiplication over the Integers

Abstract

Strassen showed that two 2x2 matrices can be multiplied with 7 multiplications instead of 8. Applied recursively, his algorithm multiplies two nxn matrices with O(n^2.807) multiplications, beating the naive O(n^3). The best known 3x3 recursive matrix multiplication algorithm uses 23 multiplications O(n^2.854). The best published lower bound of 21 (on algorithms with integer constants) leaves room for an algorithm with O(n^2.771) multiplications, and thus does not rule out the possibility of an algorithm that would beat Strassen's. We prove a lower bound of 22 multiplications for any 3x3 recursive algorithm with integer constants, proving that no such algorithm can do better than O(n^2.814) multiplications, and eliminating the possibility of a 3x3 algorithm that beats Strassen's 2x2 method. The proof builds on a recent decomposition method from Wang, who approached the problem by turning it into 496 subproblems. We provide exact solutions for 359 of them. The proof is in Lean; verification requires auditing only a few short files. The Lean formalization directly encodes statements about the limitations of recursive algorithms for matrix multiplication, as opposed to just a statement about the rank of the problem.