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?
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