In Strassen's matrix multiplication, we state one strange ( at least to me) fact that matrix multiplication of two 2 x 2 takes 7 multiplication.
Question : How to prove that it is impossible to multiply two 2 x 2 matrices in 6 multiplications?
Please note that matrices are over integers.
This is a classical result of Winograd: On multiplication of 2x2 matrices.
Strassen showed that the exponent of matrix multiplication is the same as the exponent of the tensor rank of matrix multiplication tensors: the algebraic complexity of $n\times n$ matrix multiplication is $O(n^\alpha)$ iff the tensor rank of $\langle n,n,n \rangle$ (the matrix multiplication tensor corresponding to the multiplication of two $n\times n$ matrices) is $O(n^\alpha)$. Strassen's algorithm uses the easy direction to deduce an $O(n^{\log_27})$ from the upper bound $R(\langle 2,2,2 \rangle) \leq 7$.
Winograd's result implies that $R(\langle 2,2,2 \rangle)=7$. Landsberg showed that the border rank of $\langle 2,2,2 \rangle$ is also 7, and Bläser et al. recently extended that to support rank and border support rank. Border rank and support rank are weaker (=smaller) notions of rank that have been used (in the case of border rank) or proposed (in the case of support rank) in the fast matrix multiplication algorithms.
You can find the result at:
S.Winograd, On multiplication of 2×2 matrices, Linear Algebra and Appl. 4 (1971), 381–388, MR0297115 (45:6173).
