Product of Sums of Four Squares/Proof 1
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Theorem
Let $a, b, c, d, w, x, y, z$ be numbers.
Then:
\(\ds \) | \(\) | \(\ds \left({a^2 + b^2 + c^2 + d^2}\right) \left({w^2 + x^2 + y^2 + z^2}\right)\) | ||||||||||||
\(\ds =\) | \(\) | \(\ds \left({a w + b x + c y + d z}\right)^2\) | ||||||||||||
\(\ds \) | \(+\) | \(\ds \left({a x - b w + c z - d y}\right)^2\) | ||||||||||||
\(\ds \) | \(+\) | \(\ds \left({a y - b z - c w + d x}\right)^2\) | ||||||||||||
\(\ds \) | \(+\) | \(\ds \left({a z + b y - c x - d w}\right)^2\) |
Proof
Taking each of the squares on the right hand side and multiplying them out in turn:
\(\ds \paren {a w + b x + c y + d z}^2\) | \(=\) | \(\ds a^2 w^2 + b^2 x^2 + c^2 y^2 + d^2 z^2\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 2 \paren {a b w x + a c w y + a d w z + b c x y + b d x z + c d y z}\) |
\(\ds \paren {a x - b w + c z - d y}^2\) | \(=\) | \(\ds a^2 x^2 + b^2 w^2 + c^2 z^2 + d^2 y^2\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 2 \paren {-a b w x + a c x z - a d x y - b c w z + b d w y - c d y z}\) |
\(\ds \paren {a y - b z - c w + d x}^2\) | \(=\) | \(\ds a^2 y^2 + b^2 z^2 + c^2 w^2 + d^2 x^2\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 2 \paren {-a b y z - a c w y + a d x y + b c w z - b d x z - c d w x}\) |
\(\ds \paren {a z + b y - c x - d w}^2\) | \(=\) | \(\ds a^2 z^2 + b^2 y^2 + c^2 x^2 + d^2 w^2\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 2 \paren {a b y z - a c x z - a d w z - b c x y - b d w y + c d w x}\) |
All the non-square terms cancel out with each other, leaving:
\(\ds \) | \(\) | \(\ds \paren {a w + b x + c y + d z}^2\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {a x - b w + c z - d y}^2\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {a y - b z - c w + d x}^2\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {a z + b y - c x - d w}^2\) | |||||||||||
\(\ds \) | \(=\) | \(\ds a^2 w^2 + b^2 x^2 + c^2 y^2 + d^2 z^2\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds a^2 x^2 + b^2 w^2 + c^2 z^2 + d^2 y^2\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds a^2 y^2 + b^2 z^2 + c^2 w^2 + d^2 x^2\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds a^2 z^2 + b^2 y^2 + c^2 x^2 + d^2 w^2\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a^2 + b^2 + c^2 + d^2} \paren {w^2 + x^2 + y^2 + z^2}\) |
$\blacksquare$