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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1answer
48 views

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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1answer
90 views

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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148 views

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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92 views

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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258 views

Calculation of the Pfaffian of a matrix

I have a set of $N$ numbers $\lbrace \lambda_i\rbrace_{i\in[1,N]}$ that belong to $[0,2\pi[$ and a real number $L$ and I am trying to evaluate the following Pfaffian expression. $$\mathrm{Pf}\left(\...
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71 views

Pfaffian of the antisymmetric matrix whose all upper diagonal entries are 1

What is the pfaffian of $n\times n$ antisymmetric matrix whose all upper diagonal entries are 1? Is there any method to compute?
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198 views

Numerical calculation of pfaffian

I tried to calculate numerical the pfaffian of a skew symmetric matrix by the recursive definition (from Wikipedia): $$ \text{pf}\left(A\right) = \sum_{j=2}^{2n}\left(-1\right)^{j}a_{1j}\text{pf}\...
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216 views

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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373 views

Proof of result involving Pfaffian of a matrix

Show that $$\text{Pf} MAM^T = \text{det}M \cdot \text{Pf} A$$ for any matrix $M$ and antisymmetric $A$. Attempt: $$\text{Pf} MAM^T = \frac{1}{2^N N!} \epsilon_{\alpha_1 \dots \alpha_{2N}} (MAM^T)_{\...
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Calculate Pfaffian of a special 2x2 block matrix

I have a $2\times2$ matrix $$ M= \begin{pmatrix} A & -1\\ 1 & B \\ \end{pmatrix}. $$ Here $A$ and $B$ are skew matrix, the matrix dimension is $L$. Is there a quick way to calculate the ...
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168 views

How to calculate a determinant of a general 2x2 block skew matrix?

I would like to calculate the determinant of a $2\times2$ block skew matrix: $$ \begin{pmatrix} A & B^T \\ -B & D \\ \end{pmatrix} $$ with $A^T=-A$ and $D^T=...
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Is There a Basis Free Definition of the Pfaffian

$\DeclareMathOperator{\pf}{pf}$ I recently came across a delightful fact that: The determinant of a $2n\times 2n$ skew-symmetric matrix is a the square of a certain polynomial called the pfaffian. I ...
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Pfaffian of skew symmetric block matrix

I am trying to prove that if A is a $m x m$ real matrix and $B=\begin{bmatrix}0 & A\\-A^t & 0\end{bmatrix}$, then Pfaffian(B)= $(-1)^{m(m-1)/2}$ det(A). I know that Pfaffian$(B)^2=$det(B), so ...
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Hypermatrices, hyperdeterminants and Grassmannians.

Let $Gr(k,n)$ the Grassmannian manifold of the $k$-planes in $\mathbb{C}^n$ and consider the Plucker embedding $\pi: Gr(k,n) \to \mathbb{P}(\Lambda^k \mathbb{C}^n)$. Let $A$ be the set of $n \times n$ ...
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FKT algorithm and adjacency matrices

The Wikipedia article on the FKT algorithm says that one finds the number of perfect matchings in an undirected planar graph $G$ as follows. Find a graph $H$ that is a directed version of $G$, such ...
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The close form expression of a Pfaffian

Recall Schur's Pfaffian identity: $$ \mathrm{Pf}\left(\frac{x_j-x_i}{x_j+x_i}\right)_{1\le i,j\le 2n} = \prod_{1\le i<j \le 2n}\frac{x_j-x_i}{x_j+x_i}. $$ Here $x_1,x_2\cdots x_{2n}$ are $2n$ ...
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Pfaffian And Determinant

I am working in tilings using Pfaffian. There is a basic property, namely: Let $ B$ be a $n\times n$ matrix. Let $$ A = \begin{pmatrix} 0 & B\\ -B^T & 0 \end{pmatrix}$$ then $$\...
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614 views

Simple/Concise proof of Muir's Identity

I am not a Math student and I am having trouble finding some small proof for the Muir's identity. Even a slightly lengthy but easy to understand proof would be helpful. Muir's Identity $$\det(A)= (\...
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Existence of the Pfaffian?

Consider a square skew-symmetric $n\times n$ matrix $A$. We know that $\det(A)=\det(A^T)=(-1)^n\det(A)$, so if $n$ is odd, the determinant vanishes. If $n$ is even, my book claims that the ...
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Pfaffian properties

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-...