Binet-Cauchy Identity/Proof 2

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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

This is a special case of the Cauchy-Binet Formula:

$\displaystyle \det \left({\mathbf A \mathbf B}\right) = \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \cdots \mathop < j_m \le n} \det \left({\mathbf A_{j_1 j_2 \ldots j_m}}\right) \det \left({\mathbf B_{j_1 j_2 \ldots j_m}}\right)$

where:

$\mathbf A$ is an $m \times n$ matrix
$\mathbf B$ is an $n \times m$ matrix.
For $1 \le j_1, j_2, \ldots, j_m \le n$:
$\mathbf A_{j_1 j_2 \ldots j_m}$ denotes the $m \times m$ matrix consisting of columns $j_1, j_2, \ldots, j_m$ of $\mathbf A$.
$\mathbf B_{j_1 j_2 \ldots j_m}$ denotes the $m \times m$ matrix consisting of rows $j_1, j_2, \ldots, j_m$ of $\mathbf B$.


In this case $m = 2$, giving:

$\displaystyle \det \left({\mathbf A \mathbf B}\right) = \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop \le n} \det \left({\mathbf A_{j_1 j_2} }\right) \det \left({\mathbf B_{j_1 j_2} }\right)$



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 $1812$.


Sources