By Blaser M.

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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. [Lad76] J. Laderman. A noncommutative algorithm for multiplying 3 × 3–matrices using 23 multiplications. Bull. Amer. Math. , 82:180–182, 1976. [Pan80] Victor Ya. Pan. New fast algorithms for matrix multiplication. SIAM J. Comput, 9:321–342, 1980. [Sch37] Arnold Scholz. Aufgabe 253. Mathematiker-Vereinigung, 47:41–42, 1937.

Then we still compute b1 . We can kill another β product by substituting b1 as above. After this, we still compute a0 b0 , which needs one product. However, we can approximate the tensor above by tensors of rank two. Let 1 1 t(ǫ) = (1, ǫ) ⊗ (1, ǫ) ⊗ (0, ) + (1, 0) ⊗ (1, 0) ⊗ (1, − ) ǫ ǫ t(ǫ) obviously has rank two for every ǫ > 0. The slices of t(ǫ) are 1 0 0 0 0 1 31 1 ǫ Thus t(ǫ) → t if ǫ → 0. Bini, Capovani, Lotti and Romani [BCLR79] used this effect to design better matrix multiplication algorithms.

When we replace a1 by − a0 , we kill one product. We still compute a0 b0 and β =0 α − a0 b0 + a0 b1 . Next, set a0 = 1, b0 = 0. Then we still compute b1 . We can kill another β product by substituting b1 as above. After this, we still compute a0 b0 , which needs one product. However, we can approximate the tensor above by tensors of rank two. Let 1 1 t(ǫ) = (1, ǫ) ⊗ (1, ǫ) ⊗ (0, ) + (1, 0) ⊗ (1, 0) ⊗ (1, − ) ǫ ǫ t(ǫ) obviously has rank two for every ǫ > 0. The slices of t(ǫ) are 1 0 0 0 0 1 31 1 ǫ Thus t(ǫ) → t if ǫ → 0.