Lagrange's Identity

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\(\ds \paren {\sum_{k \mathop = 1}^n {a_k}^2} \paren {\sum_{k \mathop = 1}^n {b_k}^2} - \paren {\sum_{k \mathop = 1}^n a_k b_k}^2\) \(=\) \(\ds \sum_{i \mathop = 1}^{n - 1} \sum_{j \mathop = i + 1}^n \paren {a_i b_j - a_j b_i}^2\)
\(\ds \) \(=\) \(\ds \frac 1 2 \sum_{i \mathop = 1}^n \sum_{\substack {1 \mathop \le j \mathop \le n \\ j \mathop \ne i} } \paren {a_i b_j - a_j b_i}^2\)


A special case of the Binet-Cauchy Identity.


Source of Name

This entry was named for Joseph Louis Lagrange.