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Given a $2n\times 2n$ real skew-symmetric matrix $A$, its Pfaffian $\mathrm{Pf}$ is defined as: $ \mathrm{Pf}(A) = \frac1{2^n n!}\sum_{\sigma\in S_{2n}}\mathrm{sgn}(\sigma)\prod_{i=1}^n A_{\sigma(2i-1),\sigma(2i)} $ where $S_{2n}$ is the symmetric group, and $\mathrm{sgn}(\sigma)$ is the signature of the permutation $\sigma$.

I'm looking for general properties of Pfaffians, which go beyond the ones stated in its wiki entry. More specifically, I'd like to know general conditions for the equality between the Pfaffian of a convex sum, and the convex sum of Pfaffians. In plain math, let $A$ and $B$ be $2n\times 2n$ real skew-symmetric matrices, and $\lambda \in [0,1]$. What are the conditions for the equality $\mathrm{Pf}((1-\lambda)A +\lambda \;B) = (1-\lambda)\;\mathrm{Pf}(A) +\lambda\; \mathrm{Pf}(B)$ to hold?

A general reference on Pfaffians would also be very appreciated.

Thanks,

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    @percusse I'll definitely do it, as soon as I get a more complete answer to the question. Up to now I only have special cases -- which are enough for my purposes, but not the whole thing...2011-12-05

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