Here, we describe a remarkable algorithm for matrix multiplication discovered by Strassen, [62].
Let

and

be block matrices of size n = 2m, where all blocks are of size m × m.
(a) Let
D1 = (A1 + A4) (B1 + B4),
D2 = (A1 - A3) (B1 + B2).
D3 = (A2 - A4) (B3 + B4),
D4 = (A1 + A2) Bi,
D5 = (A3 + A4) B1,
D6 = A4(B1 - B3),
D7 = A1(B2 - B2).
Show that
C1 = D1 + D3 - D4 - D6.
C2 -D4 + D7.
C3 = D5 - D6,
C4 = D1 - D2 - D5 + D7.
(b) How many arithmetic operations are required when A and B are 2 × 2 matrices? How does this compare with the usual method of multiplying 2×2 matrices?
(c) In the general case, suppose we use standard matrix multiplication for the matrix products in
D1.........D7. Prove that Strassen's method is faster than the direct algorithm for computing AB by a factor of ‰ˆ7/8.
(d) When A and B have size n × n with n = 2r, we can recursively apply Strassen's method to multiply the 2r-1 × 2r-l blocks A. B. Prove that the resulting algorithm requires a total of 7ʹ= nlog2 7= n280735 multiplications and
6(7r-1 - 4r-1) additions/subtractions, versus n3 multiplications and n3 - n2 ‰ˆ n3 additions for the ordinary matrix multiplication algorithm. How much faster is Strassen's method when n =210?225?2100?
(e) How might you proceed if the size of the matrices does not happen to be a power of 2? Further developments of these ideas can be found in [11.32].

A -(A A2 B1 B2 .).