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Below is proof of the Cauchy-Binet Formula found on Wikipedia.

Why are the last two equations equal?

Let $\textbf{A}$ be a $m\times n$ matrix and $\textbf{B}$ be a $n\times m$ matrix.

$\begin{align*} \displaystyle \det \left({\mathbf A \mathbf B}\right)&=\sum_{1 \mathop \le l_1, \mathop \ldots \mathop , l_m \mathop \le m} \epsilon \left({l_1, \ldots, l_m}\right) \left({\sum_{k \mathop = 1}^n a_{1 k} b_{k l_1} }\right) \cdots \left({\sum_{k \mathop = 1}^n a_{m k} b_{k l_m} }\right)\\ &=\sum_{1 \mathop \le k_1, \mathop \ldots \mathop , k_m \mathop \le n} a_{1 k_1} \cdots a_{m k_m} \sum_{1 \mathop \le l_1, \mathop \ldots \mathop , l_m \mathop \le m} \epsilon \left({l_1, \ldots, l_m}\right) b_{k_1 l_1} \cdots b_{k_m l_m}\\ &=\sum_{1 \mathop \le k_1, \mathop \ldots \mathop , k_m \mathop \le n} a_{1 k_1} \cdots a_{m k_m} \det \left({\mathbf B_{k_1 \cdots k_m} }\right)\\ &=\sum_{1 \mathop \le k_1, \mathop \ldots \mathop , k_m \mathop \le n} \epsilon \left({k_1, \ldots, k_m}\right)a_{1 k_1} \cdots a_{m k_m} \det \left({\mathbf B_{j_1 \cdots j_m} }\right) \tag{1}\\ &=\sum_{1 \mathop \le j_1 \mathop \le j_2 \mathop \le \cdots \mathop \le j_m \le n} \det \left({\mathbf A_{j_1 \cdots j_m} }\right) \det \left({\mathbf B_{j_1 \cdots j_m} }\right) \tag{2}\\ \end{align*}$

This is how far I got starting with equation $(2)$... $\begin{equation} \sum_{1 \mathop \le j_1 \mathop \le j_2 \mathop \le \cdots \mathop \le j_m \le n} \det \left({\mathbf A_{j_1 \cdots j_m} }\right) \det \left({\mathbf B_{j_1 \cdots j_m} }\right) = \sum_{1 \mathop \le j_1 \mathop \le j_2 \mathop \le \cdots \mathop \le j_m \le n} (\sum_{1\leq k_1,...,\ k_m\leq n}\epsilon(k_1, k_2, ...,k_m)a_{j_1,k_1}a_{j_2,k_2}...a_{j_m,k_m}) \det \left({\mathbf B_{j_1 \cdots j_m} }\right) \end{equation}$

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1 Answer 1

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The last $=$ just factors out $\det A_{j_1\cdots j_m}$, defined in terms of the Levi-Civita symbol.

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  • $\begingroup$ Why does the index chain on the lower limit of the sum then? The index for equation $(1)$ is $1\leq k_1, k_2, ..., k_m\leq n$ and for the second equation is $1\leq j_1 \leq j_2 \leq ... \leq j_m \leq n$. How does it go from there to there? $\endgroup$
    – W. G.
    May 29, 2019 at 14:02
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    $\begingroup$ I added some more to the problem rewriting the determinant with the Levi-Civita symbol. $\endgroup$
    – W. G.
    May 29, 2019 at 14:54

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