Lower bounds for matrix product

We prove lower bounds on the number of product gates in bilinear and quadratic circuits that compute the product of two n × n matrices over finite fields. In particular we obtain the following results: 1. We show that the number of product gates in any bilinear (or quadratic) circuit that computes the product of two n × n matrices over GF(2) is at least 3n² - o(n²). 2. We show that the number of product gates in any bilinear circuit that computes the product of two n × n matrices over GF(q) is at least (2.5 + 1.5/q³-1)n²-o(n²). These results improve the former results of [N. H. Bshouty, SIAM J. Comput., 18 (1989), pp. 759-765; M. Bläser, Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 1999, pp. 45-50], who proved lower bounds of 2.5n² - o(n²).