Complex matrix operations question

Question

If we have $4$ Real matrices $A,B,C,D$, is it possible to show that $(A+Bi)(C+Di) = E+Fi$ in $3$ nxn matrix multiplications?

Do you mean, to determine $E, F$ in terms of $A, B, C, D$? – Travis Willse Dec 16 '14 at 17:24 — Travis Willse, Dec 16 '14 at 17:24

score 1 · Accepted Answer · answered Dec 16 '14 at 17:30

Same as for real numbers $a,b,c,d$:

$$(a+ib)(c+id)=(ac-bd)+i(ad+bc)$$

And with $(a+b)(c+d)=(ac+bd)+(ad+bc)=t$ and $u=ac$, $v=bd$, you have then that

$$(a+ib)(c+id)=(u-v)+i(t-u-v)$$

We didn't use commutativity (except for $i$, but it won't hurt as it's also a scalar in your case), hence it works for matrices too.

This is just a variation on Karatsuba algorithm.

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