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We will try to make use of this in the following. 2. Let t ∈ K k×m×n and t′ ∈ K k ×m ×n . ′ ′ 1. t is called a restriction of t′ if there are homomorphisms α : K k → K k , β : K m → K m , ′ and γ : K n → K n such that t = (α ⊗ β ⊗ γ)t′ . We write t ≤ t′ . 2. t and t′ are isomorphic if α, β, γ are isomorphisms (t ∼ = t′ ). In the following, r denotes the tensor in K r×r×r that has a 1 in the positions (ρ, ρ, ρ), 1 ≤ ρ ≤ r, and 0s elsewhere. This tensor corresponds to the r bilinear forms xρ yρ , 1 ≤ ρ ≤ r (r independent products).

Letters, 8:234–235, 1979. [BCS97] Peter B¨ urgisser, Michael Clausen, and M. Amin Shokrollahi. Algebraic Complexity Theory. Springer, 1997. [Bl¨a03] Markus Bl¨aser. On the complexity of the multiplication of matrices of small formats. J. Complexity, 19:43–60, 2003. [Bra39] A. T. Brauer. On addition chains. Bulletin of the American Mathematical Society, 45:736–739, 1939. [Bsh95] Nader H. Bshouty. On the additive complexity of 2 × 2-matrix multiplication. Inform. Proc. Letters, 56(6):329–336, 1995.

This kills two products and we still compute z11 , z21 . Consider the following five products: p1 = (x12 + ǫx22 )y21 p2 = x11 (y11 + ǫy12 ) p3 = x12 (y12 + y21 + ǫy22 ) p4 = (x11 + x12 + ǫx21 )y11 p5 = (x12 + ǫx21 )(y11 + ǫy22 ) We have ǫz11 = ǫp1 + ǫp2 + O(ǫ2 ) ǫz12 = p2 − p4 + p5 + O(ǫ2 ) ǫz21 = p1 − p3 + p5 + O(ǫ2 ) Now we take a second copy of the partial matrix multiplication above, with new variables. With these two copies, we can multiply 2 × 2-matrices with 2 × 3-matrices (by identifying some of the variables in the copy).

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