On the complexity of matrix product

We prove a lower bound of Ω(m² log m) for the size of any arithmetic circuit for the product of two matrices, over the real or complex numbers, as long as the circuit doesn't use products with field elements of absolute value larger than 1 (where m × m is the size of each matrix). That is, our lower bound is super-linear in the number of inputs and is applied for circuits that use addition gates, product gates and products with field elements of absolute value up to 1. More generally, for any c = c(m) ≥ 1, we obtain a lower bound of Ω(m²log_2c m) for the size of any arithmetic circuit for the product of two matrices (over the real or complex numbers), as long as the circuit doesn't use products with field elements of absolute value larger than c. We also prove size-depth tradeoffs for such circuits.