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I recently proved for homework the following identity on $\mathbb{C}$: if $a_1, \ldots , a_n, b_1, \ldots, b_n\in\mathbb{C}$, then $$ \left|\sum_{i=1}^na_ib_i\right|^2 = \left(\sum_{i=1}^n|a_i|^2\right)\left(\sum_{i=1}^n|b_i|^2\right) - \sum_{1\leq i<j\leq n} |a_i\overline{b_j}-a_j\overline{b_i}|^2. $$ This identity is called Lagrange's identity. I was wondering what are some applications of this identity. I know that one can infer Cauchy's inequality, but I was wondering if there were any other uses of it.

Thanks!!

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