Binet-Cauchy Identity

Theorem

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

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:

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

Hence the result.

$\blacksquare$

Note

This is in fact a special case of the Cauchy-Binet Formula.

Source of Name

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

It is also known as Binet's formula.

Sources

• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.3$: Exercise $30$