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Questions tagged [pfaffian]

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

The determinant of a skew-symmetric matrix is a square of some polynomial in coefficients of the matrix, $\det(A)=\operatorname{pf}(A)^2$. This polynomial is called Pfaffian.

Explicitly $$ \operatorname{pf}(A)= \frac{1}{2^n n!}\sum_{\sigma \in S_{2n}}\operatorname{sgn}(\sigma) \prod_{i = 1}^n A_{\sigma(2i-1),\sigma(2i)} $$

Pfaffians are applied for counting perfect matching in graphs (domino tilings &c).

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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 determinant is the square of a polynomial function of the…
Potato
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Pfaffian properties

Given a $2n\times 2n$ real skew-symmetric matrix $A$, its Pfaffian $\mathrm{Pf}$ is defined as: $$ \mathrm{Pf}(A) = \frac1{2^n n!}\sum_{\sigma\in S_{2n}}\mathrm{sgn}(\sigma)\prod_{i=1}^n A_{\sigma(2i-1),\sigma(2i)} $$ where $S_{2n}$ is the…
Fernando
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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 was looking for a "conceptual proof" of the above.…
caffeinemachine
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Prove that the Pfaffian satisfies $\text{Pf}(MAM^T)=\det(M)\text{Pf}(A)$

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)_{\alpha_1 \alpha_2} \dots (MAM^T)_{\alpha_{2N-1}…
CAF
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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 every skew-symmetric matrix A . As for determinants…
user395911
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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
Isidore Seville
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Gauss-Bonnet Theorem in dimension four

I've read that the generalized Gauss-Bonnet theorem states that $$\int\limits_{M}Pf(\Omega)=(2\pi)^n\chi(M)$$ where, $M$ is a 2n-dimensional compact orientable Riemannian manifold without boundary $\Omega$ is the curvature form and $Pf(\Omega)$ is…
DiegoMath
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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 it is known that the determinant of an…
UglyMousanova19
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Prove that $\det\begin{pmatrix}A&B\\-B&A\end{pmatrix}$ is a sum of squares of polynomials

As discussed for example in this question, given any pair of real squared matrices $A,B$ we have the identity $$|\det(A+iB)|^2 = \det\begin{pmatrix}A&B\\ -B&A\end{pmatrix}.$$ In particular, this means that $\det\begin{pmatrix}A&B\\…
glS
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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 $n \times n$ skew-symmetric matrices and fix a natural number…
user137921
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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 $$\det(A)= (\operatorname{pf}(A))^2;$$ the identity is given in…
Vk1
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An easier evaluation of $\det\limits_{1\leqslant i,j\leqslant n}\left\{\frac{x_i-x_j}{x_i+x_j}\right\}$

I'm looking for an easier proof of the identity (attributed to K. F. W. Rohn) $$R_n(\bar{x}):=\det_{1\leqslant i,j\leqslant n}\left\{\frac{x_i-x_j}{x_i+x_j}\right\}=\prod_{i
metamorphy
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Relation between Pfaffian and determinant

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…
Reya Rose
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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(\frac{\frac{\lambda_j }{i \tan(\frac{\lambda_j…
Alexandre Krajenbrink
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Computation of the pfaffian of a particular matrix

Let $\{s_i\}_i$ be a sequence of integers such that $s_i>\sum\limits_{j=1}^{i-1}s_j$ with $s_1=1.$ For $\alpha \in (0,1)$ define the $n \times n$ antisymmetric matrix $A(n)$ by induction: $A(2)=\left(\begin{array}{cc}0 & \alpha^{s_1} \\…
SION
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