# Questions tagged [pfaffian]

For questions about Pfaffians of skew-symmetric matrices, $\det(A)=\operatorname{pf}(A)^2$.

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### Relation between Pfaffian and determinant

I know the relation connecting Pfaffian and determinant is given by: det(BAB^T)=det(B)Pf(A), For an arbitrary 2n × 2n matrix B, and A is a 2n x 2n real antisymmetric matrix. But do anybody know ...
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### Is there a generalization of Pfaffians?

For an skew-symmetric matrix $A$ (meaning $A^T=-A$), the Pfaffian is defined by the equation $(\text{Pf}\,A)^2=\det A$. It is my understanding that this is defined for anti-symmetric matrices because ...
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### Literature to get started on Pfaffian systems

I am getting into Pfaffian systems as a way to represent systems of PDEs. I was wondering what could be a good book to get started. I found some notes and lectures online, but nowhere a clear ...
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### Independent indeterminate elements

I'm trying to figure out what 'skew symmetric matrix in which the elements above the diagonal are independent indeterminates over the ring of rational integers' means. I'm kind of confused by this ...
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### A non-zero quantity associated to an invertible skew-symmetric matrix of even order.

Once again, I failed to make a concise post so feel free to skip to the emphasized parts. In the context of symplectic and contact geometry, I would like to establish the following linear algebra ...
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### Counting Lattice Paths with Pfaffian

This problem comes from Stanley's Enumerative Combinatorics Volume 1 (Problem 37, page 265). The problem statement is quite long, so I have added an image, but as a short synopsis the problem asks to ...
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### Solution vs Primitive

What is considered as a solution of the Pfaffian DE $Pdx+Qdy+Rdz=0$? If the LHS is the total differential of a function $F(x,y,z)$ then $F$ is called the primitive of the Pf DE. My book says $y=y(x)$, ...
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### diagonal map on exterior algebra

Let $F$ be a free module of rank $2m$ over a commutative ring $R$. In Buchsbaum-Eisenbud's paper from 1977 about structure theorems for free resolutions of ideals of codimension 3, they give a proof ...
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### Analogs of Cayley-Hamilton theorem for Pfaffian

The Pfaffian $\text{pf}$ is defined for a skew-symmetric matrix which is also a polynomial of matrix coefficients. One property for Pfaffian is that $\operatorname {pf} (A)^{2}=\det(A)$ holds for ...
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### Laplace expansion of Pfaffian

I am reading about Pfaffian, which can be found here https://en.wikipedia.org/wiki/Pfaffian. We know that the (general) Laplace expansion is very useful to compute determinants of matrices, and I ...
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