The product of two complex numbers in less than 3 multiplications

Question

Can someone break this for me? Why is this impossible to do with two multiplications?

Multiplication of complex numbers

If the number of multiplications necessary for calculation is considered as a measure of its complexity, and these calculations are performed using complex numbers, it is natural to ask how many real multiplications are needed to evaluate the real and imaginary parts of a complex product. The natural way to form a complex product requires four real multiplications. However, this can be done in three cases, but not in two combinations.

(a+bi)(c+di) = (ac-bd) + (ad+bc)i

a(c+d) - d(a+b) = ac - bd
 (1)      (2)

a(c+d) + c(b-a) = ad + bc
          (3)

Theorem . To evaluate the product of two complex numbers, three real multiplications are required, even if the multiplication by real constants does not count.

. , , , C _i, W _i, X _i, Y _i Z _i .

ac - bd = C₁(W₁a+X₁b+Y₁c+Z₁d)
            (W₂a+X₂b+Y₂c+Z₂d)
        + C₂(W₃a+X₃b+Y₃c+Z₃d)
            (W₄a+X₄b+Y₄c+Z₄d)
ad + bc = C₃(W₁a+X₁b+Y₁c+Z₁d)
            (W₂a+X₂b+Y₂c+Z₂d)
        + C₄(W₃a+X₃b+Y₃c+Z₃d)
            (W₄a+X₄b+Y₄c+Z₄d)

20 20 , C _i, W _i, X _i, Y _i Z _i, (i = 1,2,3,4), , ,

:

, . "40-44". http://dl.acm.org/. Proc. ACM , , . . , . . . Acm, 03 1971. . 26 2016. http://dl.acm.org/citation.cfm?doid=800157.805036.

Answer 1

, , , : " , , , ac-bd ad + bc -- ."

( : " (s) " "MNPQ (s)".)

, , , {ac-bd, ad + bc} MNPQ. , , MNPQ, - - "" MNPQ, {ac-bd, ad + bc}.

, , , , , , , , . ( , ?) , MNPQ {a, b, c, d} , "" , {ac-bd, ad + bc } - MNPQ. ( , MNPQ , , MNPQ, {a, b, c, d}, " ", , .)

, MNPQ (W₁a + X₁b + Y₁c + Z₁d) middot; (W₂a + X₂b + Y₂c + Z₂d) (W₃a + X₃b + Y₃c + Z₃d) & middot; ( W₄a + X₄b + Y₄c + Z₄d) (ac-bd ad + bc) C₁ (MNPQ) ₁ + C₂ (MNPQ) ₂ C₃ (MNPQ) ₃ + C₄ (MNPQ) ₄. , ; , W₁, X₂, C₃ .. — , , , . , , , MNPQ ac-bd, ad + bc.

Answer 2

.

, 2 , ac-bd ad + bc, 2 .

, C1, C2, C3, C4, Xi, Yi, Zi, Wi .

a ^ 2, b ^ 2, c ^ 2, d ^ 2, ab, ac, ad, bc, bd, cd , 20 20 . , C1 * W1 * W2 + C2 * W3 * W4 = 0 a ^ 2 ac-bd. , , .