It is well known that 2x2 matrices can be multiplied using just 7 (instead of the obvious 8) multiplications in the ground field (Strassen-Winograd, etc.). It is also well known that complex numbers can be multiplied using just 3 real multiplications, using what is basically Karatsuba polynomial multiplication. You can obviously combine these tricks, either way round, to multiply two complex 2x2 matrices using 21 real multiplies (and some additions etc.). Is it known whether you can do better? Is there any way to do the whole computation in 20 or fewer multiplications?

  • $\begingroup$ Look what Hopcroft, Kerr: "On Minimizing the Number of Multiplications Necessary for Matrix Multiplication" dragged in. $\endgroup$ – greybeard Jan 11 '18 at 21:20
  • $\begingroup$ I can only access the abstract from here, but it doesn't look like it tackles the complex case, which is the key element of my question. $\endgroup$ – Steve Linton Jan 13 '18 at 16:22
  • $\begingroup$ (I think I saw the full article accessibe - didn't seem legit to me: no link. If I was under the impression that it presents a result - negative or positive - I'd answer. I think this is more about the structure of than the structure of processing elements of : mathoverflow?) $\endgroup$ – greybeard Jan 13 '18 at 17:04

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