Matrix Multiplication I

These notes are based on a lecture given at the Toronto Student Seminar on February 2, 2012. The material is taken mostly from the book Algebraic Complexity Theory [ACT] and the lecture notes by Bläser and Bendun [Blä]. Starred sections are the ones I didn't have time to cover. This lecture discusses the problem of multiplying two square matrices. We will be working in the algebraic complexity model. For us, an algorithm for multiplying two n × n matrices will mean a sequence of steps, where step l is a statement of the form • t l ← r for any r ∈ R • c ij ← t p for p < l We will say that such an algorithm computes the product C = AB if at the end of the program, c ik = j a ij b jk. The running-time or complexity of the algorithm is the total number of steps, disregarding input and output steps. Our model is non-uniform. As an example, consider Strassen's algorithm for multiplying two 2 × 2 matrices, as copied form Wikipedia: