Binet-Cauchy Identity/Proof 1

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Theorem

$\displaystyle \left({\sum_{i \mathop = 1}^n a_i c_i}\right) \left({\sum_{j \mathop = 1}^n b_j d_j}\right) = \left({\sum_{i \mathop = 1}^n a_i d_i}\right) \left({\sum_{j \mathop = 1}^n b_j c_j}\right) + \sum_{1 \mathop \le i \mathop < j \mathop \le n} \left({a_i b_j - a_j b_i}\right) \left({c_i d_j - c_j d_i}\right)$

where all of the $a, b, c, d$ are elements of a commutative ring.

Thus the identity holds for $\Z, \Q, \R, \C$.


Proof

Expanding the last term:

\(\displaystyle \) \(\) \(\displaystyle \sum_{1 \mathop \le i \mathop < j \mathop \le n} \left({a_i b_j - a_j b_i}\right) \left({c_i d_j - c_j d_i}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{1 \mathop \le i \mathop < j \mathop \le n} \left({a_i c_i b_j d_j + a_j c_j b_i d_i}\right)\)
\(\displaystyle \) \(\) \(\, \displaystyle - \, \) \(\displaystyle \sum_{1 \mathop \le i \mathop < j \mathop \le n} \left({a_i d_i b_j c_j + a_j d_j b_i c_i}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{1 \mathop \le i \mathop < j \mathop \le n} \left({a_i c_i b_j d_j + a_j c_j b_i d_i}\right) + \sum_{i \mathop = 1}^n a_i c_i b_i d_i\) These new terms are the same
\(\displaystyle \) \(\) \(\, \displaystyle - \, \) \(\displaystyle \sum_{1 \mathop \le i \mathop < j \mathop \le n} \left({a_i d_i b_j c_j + a_j d_j b_i c_i}\right) - \sum_{i \mathop = 1}^n a_i d_i b_i c_i\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n a_i c_i b_j d_j - \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n a_i d_i b_j c_j\) Completing the sums
\(\displaystyle \) \(=\) \(\displaystyle \left({\sum_{i \mathop = 1}^n a_i c_i}\right) \left({\sum_{j \mathop = 1}^n b_j d_j}\right) - \left({\sum_{i \mathop = 1}^n a_i d_i}\right) \left({\sum_{j \mathop = 1}^n b_j c_j}\right)\) Factoring terms indexed by $i$ and $j$


Hence the result.

$\blacksquare$


Source of Name

This entry was named for Jacques Philippe Marie Binet and Augustin Louis Cauchy.


Historical Note

The Binet-Cauchy Identity is a special case of the Cauchy-Binet Formula, which was presented by Jacques Philippe Marie Binet and Augustin Louis Cauchy on the same day in $1812$.