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I know the relation connecting Pfaffian and determinant is given by $\det(BAB^\top)=\det(B)\operatorname{Pf}(A)$ where $B$ is an arbitrary $2n$ × $2n$ matrix and $A$ is a $2n$ x $2n$ real antisymmetric matrix.

But do anybody know the exact place where it is proved mathematically?

Semiclassical
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Reya Rose
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1 Answers1

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Pfaffian of a skew symmetric $n\times n$ matrix $A$ with $n=2m$ even is defined as $$Pf(A)=\frac{1}{2^m m!} \sum_{i_1,...,i_n} \epsilon^{i_1...i_n} a_{i_1i_2}\cdot \cdot \cdot a_{i_{n-1}i_n}, $$ where $\epsilon^{i_1...i_n}$ are Levi Civita symbols: $$\epsilon^{i_1...i_n}:= \begin{cases} +1&\text{if } (i_1,...,i_n) \text{ is an even permutation of } (1,...,n) \\ -1&\text{if } (i_1,...,i_n) \text{ is an odd permutation of } (1,...,n)\\ 0& \text{else } \end{cases}$$

As direct consequence Leibniz formula for determinant becomes $$ det(B)=\sum_{i_1,...,i_n} \epsilon^{i_1...i_n}b_{1i_1}\cdot \cdot \cdot b_{ni_n},$$ multiplied both sides by $\epsilon^{j_1...j_n}$ we have $$ \epsilon^{j_1...j_n}det(B)=\sum_{i_1,...,i_n} \epsilon^{i_1...i_n} b_{j_1i_1}b_{j_2i_2} \cdot \cdot \cdot b_{j_{n-1}i_{n-1}} b_{j_ni_n}. $$

Now recall that $(B^tAB)_{\textbf{ij}}=\sum_{k=1}^n \sum_{l=1}^n b_{k\textbf{i}} a_{kl} b_{l\textbf{j}}$ and notice that we are free to choose the name for indices $k,l$ in the sum. They will become $j_{s-1},j_{s}$ in equation below.

\begin{align} 2^m m!Pf(B^tAB)&=\sum_{i_1,...,i_n} \epsilon^{i_1...i_n} (B^tAB)_{i_1i_2}\cdot \cdot \cdot(B^tAB)_{i_{n-1}i_n}\\ &=\sum_{i_1,...,i_n} \epsilon^{i_1...i_n} \sum_{j_1,...,j_n}\big(b_{j_1i_1}a_{j_1j_2}b_{j_2i_2}\big) \cdot \cdot \cdot \big(b_{j_{n-1}i_{n-1}}a_{j_{n-1}j_n}b_{j_ni_n}\big)\\ &=\sum_{j_1,...,j_n} \sum_{i_1,...,i_n} \big[ \epsilon^{i_1...i_n} b_{j_1i_1}b_{j_2i_2} \cdot \cdot \cdot b_{j_{n-1}i_{n-1}} b_{j_ni_n}\big] (a_{j_1j_2} \cdot \cdot \cdot a_{j_{n-1}j_n})\\ &=\sum_{j_1,...,j_n} \epsilon^{j_1...j_n}det(B) (a_{j_1j_2} \cdot \cdot \cdot a_{j_{n-1}j_n})=det(B) 2^m m! Pf(A) \end{align}

This proof is taken from Michael Spivaks Comprehensive Introduction to Differential Geometry. vol 5-Publish or Perish (1999).

user3342072
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