Matrix multiplication complexity

Two n×n matrices multiplied by the schoolbook method require n³ multiplications. For large n that dominates training and inference cost, so the exponent — call it ω — is what researchers try to shrink. The theoretical floor is ω = 2 (you can't do better than reading all the entries); current algorithms sit around 2.37. Each small drop in ω compounds enormously at scale. The hard part is that lower-ω algorithms trade multiplications for additions in ways that become numerically unstable or only win at sizes no real machine can run.