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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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Here are a couple of applications of the identity :

1) In a paper by Stan Chirita, he discusses uniqueness and continuous data dependence questions on initial boundary value problems in thermoelasticity. The Lagrange identity is used to obtain results for bounded domains and exterior domains.

2) The Lagrange identity has been applied to Weyl-Titchmarsh theory (in the theory of Weyl disks and square summable solutions), and has a generalisation in symplectic systems. It can be shown that under an Atkinson condition, Weyl disks and square summable solutions remain valid without any change to the system.

I can provide links to these documents if you like.

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