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.


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    $\begingroup$ I found some literature on Pfaffian sums. Check for instance the series of three articles by M. Ishikawa, S. Okada and M. Wakayama entitled Applications of Minor-Summation Formula. The answer to the question I posed seems to not be explicitly given there, but the articles, specially the second one, contain all sort of properties about Pfaffians. $\endgroup$ – Fernando Dec 1 '11 at 14:39
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    $\begingroup$ You might consider answering your own question with the update and accept it. (Completely for archive reasons ;) $\endgroup$ – user13838 Dec 3 '11 at 4:22
  • $\begingroup$ @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... $\endgroup$ – Fernando Dec 5 '11 at 11:42

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