Strassen's Algorithm Made (Somewhat) More Natural: A Pedagogical Remark

Strassen's 1969 algorithm for fast matrix multiplication 2] is based on the possibility to multiply two 2 2 matrices A and B by using 7 multiplications instead of the usual 8. The corresponding formulas are an important part of any algorithms course, but, unfortunately, even in the best textbook expositions (see, e.g., 1]), they look very ad hoc. In this paper, we show that the use of natural symmetries can make these formulas more natural. Outline. The goal of this paper is to show that the use of symmetries can make Strassen's formulas for multiplying two 2 2 matrices A and B in 7 multiplications more natural. To achieve this goal, we will rst describe two relevant symmetries: the rst one is more straightforward, the second one is slightly more implicit. Then, we use these symmetries to select 7 combinations of matrix elements. Finally, we use the same symmetries to pair the combinations corresponding to A and B with each other and thus, to come up with Strassen's formulas. A = a 11 a 12 a 21 a 22 describes a linear transformation of a 2-dimensional space into itself; the product of two matrices corresponds to the composition of two linear transformations. In this interpretation, the elements of the matrix describe the coordinates of the results Ae 1 and Ae 2 of applying this transformation to the unit vectors e 1 = (1; 0) and e 2 = (0; 1) corresponding to the natural axes: Ae 1 = a 11 e 1 +a 21 e 2 , and Ae 2 = a 12 e 1 + a 22 e 2. From this geometric viewpoint what matters is the transformation itself, while the order in which we describe the axes is irrelevant. If we change this order, we get new unit vectors e 0 i = e (i) ; where (1) = 2 and (2) = 1. In the new axes, the same transformation is represented by a new matrix A 0 = (A) with a 0 ij = a (i)(j) : Similarly, the transformation corresponding to the matrix B is represented, in the new axes, by the similarly permuted matrix B 0 = (B) with b 0 ij = b (i)(j) , and the product matrix C = AB (corresponding to the composition of the two transformations) takes the form C 0 = (C) with c 0 ij = c …