Sum of Squares of Complex Moduli of Sum and Differences of Complex Numbers

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Theorem

Let $\alpha, \beta \in \C$ be complex numbers.

Then:

$\left\lvert{\alpha + \beta}\right\rvert^2 + \left\lvert{\alpha - \beta}\right\rvert^2 = 2 \left\lvert{\alpha}\right\rvert^2 + 2 \left\lvert{\beta}\right\rvert^2$


Proof

Let:

$\alpha = x_1 + i y_1$
$\beta = x_2 + i y_2$

Then:

\(\displaystyle \) \(\) \(\displaystyle \left\lvert{\alpha + \beta}\right\rvert^2 + \left\lvert{\alpha - \beta}\right\rvert^2\)
\(\displaystyle \) \(=\) \(\displaystyle \left\lvert{\left({x_1 + i y_1}\right) + \left({x_2 + i y_2}\right)}\right\rvert^2 + \left\lvert{\left({x_1 + i y_1}\right) - \left({x_2 + i y_2}\right)}\right\rvert^2\) Definition of $\alpha$ and $\beta$
\(\displaystyle \) \(=\) \(\displaystyle \left\lvert{\left({x_1 + x_2}\right) + i \left({y_1 + y_2}\right)}\right\rvert^2 + \left\lvert{\left({x_1 - x_2}\right) + i \left({y_1 - y_2}\right)}\right\rvert^2\) Definition of Complex Addition, Definition of Complex Subtraction
\(\displaystyle \) \(=\) \(\displaystyle \left({x_1 + x_2}\right)^2 + \left({y_1 + y_2}\right)^2 + \left({x_1 - x_2}\right)^2 + \left({y_1 - y_2}\right)^2\) Definition of Complex Modulus
\(\displaystyle \) \(=\) \(\displaystyle {x_1}^2 + 2 x_1 x_2 + {x_2}^2 + {y_1}^2 + 2 y_1 y_2 + {y_2}^2 + {x_1}^2 - 2 x_1 x_2 + {x_2}^2 + {y_1}^2 - 2 y_1 y_2 + {y_2}^2\) Square of Sum, Square of Difference
\(\displaystyle \) \(=\) \(\displaystyle 2 {x_1}^2 + 2 {x_2}^2 + 2 {y_1}^2 + 2 {y_2}^2\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle 2 \left({ {x_1}^2 + {y_1}^2}\right) + 2 \left({ {x_2}^2 + {y_2}^2}\right)\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle 2 \left\lvert{x_1 + i y_1}\right\rvert^2 + 2 \left\lvert{x_2 + i y_2}\right\rvert^2\) Definition of Complex Modulus
\(\displaystyle \) \(=\) \(\displaystyle 2 \left\lvert{\alpha}\right\rvert^2 + 2 \left\lvert{\beta}\right\rvert^2\) Definition of $\alpha$ and $\beta$

$\blacksquare$


Sources