On the complexity of matrix product

Our main result is 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 does not 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 superlinear 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. We also prove size-depth tradeoffs for such circuits: We show that if a circuit, as above, is of depth d, then its size is Ω(m^2+1/O(d)).