Lagrange's Identity
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Theorem
\(\ds \paren {\sum_{k \mathop = 1}^n {a_k}^2} \paren {\sum_{k \mathop = 1}^n {b_k}^2} - \paren {\sum_{k \mathop = 1}^n a_k b_k}^2\) | \(=\) | \(\ds \sum_{i \mathop = 1}^{n - 1} \sum_{j \mathop = i + 1}^n \paren {a_i b_j - a_j b_i}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \sum_{i \mathop = 1}^n \sum_{\substack {1 \mathop \le j \mathop \le n \\ j \mathop \ne i} } \paren {a_i b_j - a_j b_i}^2\) |
Proof
A special case of the Binet-Cauchy Identity.
$\blacksquare$
Source of Name
This entry was named for Joseph Louis Lagrange.