Euler's Four-Square Identity

Revision as of 08:45, 29 March 2019 by Lilcritters (talk | contribs) (Proof)

Identity

The Four-Square Identity, credited to Leonhard Euler, states that for any eight complex numbers $x_1,x_2, x_3, x_4, y_1, y_2, y_3, y_4$, we must have \[(x_1^2+ x_2^2 + x_3 ^2 + x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2)\] \[=(x_1y_1+x_2y_2+x_3y_3+x_4y_4)^2\] \[+(x_1y_2-x_2y_1+x_3y_4-x_4y_3)^2\] \[+(x_1y_3-x_3y_1+x_4y_2-x_2y_4)^2\] \[+(x_1y_4-x_4y_1 + x_2y_3 - x_3y_2)^2.\]

Proof

First, let us expand the left-hand side of the identity: \[(x_1^2+x_2^2+x_3^2+x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2) = x_1^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_2^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_3^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_4^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2)\] \[= x_1^2 y_1^2 + x_1^2 y_2^2 + x_1^2 y_3^2 + x_1^2 y_4^2 + x_2^2 y_1^2 + x_2^2 y_2^2 + x_2^2 y_3^2 + x_2^2 y_4^2 + x_3^2 y_1^2 + x_3^2 y_2^2 + x_3^2 y_3^2 + x_3^2 y_4^2 + x_4^2 y_1^2 + x_4^2 y_2^2 + x_4^2 y_3^2 + x_4^2 y_4^2.\] Thus, we have \[(x_1^2+x_2^2+x_3^2+x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2) = x_1^2 y_1^2 + x_1^2 y_2^2 + x_1^2 y_3^2 + x_1^2 y_4^2\] \[+ x_2^2 y_1^2 + x_2^2 y_2^2 + x_2^2 y_3^2 + x_2^2 y_4^2\] \[+ x_3^2 y_1^2 + x_3^2 y_2^2 + x_3^2 y_3^2 + x_3^2 y_4^2\] \[+ x_4^2 y_1^2 + x_4^2 y_2^2 + x_4^2 y_3^2 + x_4^2 y_4^2. \text{           (1)}\]

Now, let us expand the first square of the right-hand side of the identity, $(x_1y_1+x_2y_2+x_3y_3+x_4y_4)^2$: \[(x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4)^2 = (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4)(x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4)\] \[= x_1y_1 (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4) + x_2 y_2 (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4) + x_3 y_3 (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4) + x_4 y_4 (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4)\] \[= x_1^2 y_1^2 + x_1x_2y_1y_2 + x_1x_3y_1y_3 + x_1x_4y_1y_4 + x_1x_2y_1y_2 + x_2^2 y_2^2 + x_2x_3y_2y_3 + x_2x_4y_2y_4 + x_1x_3y_1y_3 + x_2x_3y_2y_3 + x_3^2 y_3^2 + x_3x_4y_3y_4 + x_1x_4y_1y_4 + x_2x_4y_2y_4 + x_3x_4y_3y_4 + x_4^2 y_4^2\] \[= (x_1^2 y_1^2 + x_2^2 y_2^2 + x_3^2 y_3^2 + x_4^2 y_4^2) + (2x_1x_2y_1y_2 + 2x_1x_3y_1y_3 + 2x_1x_4y_1y_4 + 2x_2x_3y_2y_3 + 2x_2x_4y_2y_4 + 2x_3x_4y_3y_4).\] Therefore, we now also have \[(x_1y_1+x_2y_2+x_3y_3+x_4y_4)^2 = (x_1^2 y_1^2 + x_2^2 y_2^2 + x_3^2 y_3^2 + x_4^2 y_4^2) + (2x_1x_2y_1y_2 + 2x_1x_3y_1y_3 + 2x_1x_4y_1y_4 + 2x_2x_3y_2y_3 + 2x_2x_4y_2y_4 + 2x_3x_4y_3y_4). \text{  (2)}\]

Next, let us expand the second square of the right-hand side of the identity, $(x_1y_2-x_2y_1+x_3y_4-x_4y_3)^2$: \[(x_1y_2-x_2y_1+x_3y_4-x_4y_3)^2 = (x_1y_2-x_2y_1+x_3y_4-x_4y_3)(x_1y_2-x_2y_1+x_3y_4-x_4y_3)\] \[= x_1y_2 \cdot (x_1y_2-x_2y_1+x_3y_4-x_4y_3) - x_2y_1 \cdot (x_1y_2-x_2y_1+x_3y_4-x_4y_3) + x_3y_4 \cdot (x_1y_2-x_2y_1+x_3y_4-x_4y_3) - x_4y_3 \cdot (x_1y_2-x_2y_1+x_3y_4-x_4y_3)\] \[= x_1^2 y_2^2 - x_1x_2y_1y_2 + x_1x_3y_2y_4 - x_1x_4y_2y_3 - x_1x_2y_1y_2 + x_2^2y_1^2 - x_2x_3y_1y_4 + x_2x_4y_1y_3 + x_1x_3y_2y_4 - x_2x_3y_1y_4 + x_3^2y_4^2 - x_3x_4y_3y_4 - x_1x_4y_2y_3 + x_2x_4y_1y_3 - x_3x_4y_3y_4 + x_4^2 y_3^2\] \[= (x_1^2 y_2^2 + x_2^2 y_1^2 + x_3^2 y_4^2 + x_4^2 y_3^2) + (-2x_1x_2y_1y_2 + 2x_1x_3y_2y_4 - 2x_1x_4y_2y_3 - 2x_2x_3y_1y_4 + 2x_2x_4y_1y_3 - 2x_3x_4y_3y_4).\]