Pfaffian

The Pfaffian is an analog of the determinant that is defined only for a $2n\\times 2n$ antisymmetric matrix. It is a polynomial of the polynomial ring in elements of the matrix, such that its square is equal to the determinant of the matrix.

The Pfaffian is applied in the generalized Gauss-Bonnet theorem.

Examples

$Pf\\begin{bmatrix}0&a\\\\-a&0\\end{bmatrix}=a,$

$Pf\\begin{bmatrix}0&a&b&c\\\\-a&0&d&e\\\\-b&-d&0&f\\\\-c&-e&-f&0\\end{bmatrix}=af-be+dc.$

Standard definition

Let

A=\\begin{bmatrix}0&a_{1,2}&\\ldots&a_{1,2n}\\\\-a_{1,2}&0&\\ldots&a_{2,2n}\\\\\\vdots&\\vdots&\\vdots&\\vdots\\\\-a_{2n,1}&-a_{2n,2}&\\ldots&0\\end{bmatrix}.

Let $\\Pi$ be the set of all partition of $\\{1,2,\\ldots,2n\\}$ into pairs of elements $\\alpha\\in\\Pi$ , can be represented as

\\alpha=\\{(i_{1},j_{1}),(i_{2},j_{2}),\\ldots,(i_{n},j_{n})\\}

with $i_{k}<j_{k}$ and $i_{1}<i_{2}<\\cdots<i_{n}$ , let

\\pi=\\begin{bmatrix}1&2&3&4&\\ldots&2n\\\\i_{1}&j_{1}&i_{2}&j_{2}&\\ldots&j_{n}\\end{bmatrix}

be a corresponding permutation and let us define $sgn(\\alpha)$ to be the signature of a permutation $\\pi$ ; clearly it depends only on the partition $\\alpha$ and not on the particular choice of $\\pi$ .Given a partition $\\alpha$ as above let us set $a_{\\alpha}=a_{i_{1},j_{1}}a_{i_{2},j_{2}}\\ldots a_{i_{n},j_{n}},$ then we can define the Pfaffian of $A$ as

Pf(A)=\\sum_{\\alpha\\in\\Pi}sgn(\\alpha)a_{\\alpha}.

Alternative definition

One can associate to any antisymmetric $2n\\times 2n$ matrix $A=\\{a_{ij}\\}$ a bivector: $\\omega=\\sum_{i<j}a_{ij}e_{i}\\wedge e_{j}$ in a basis $\\{e_{1},e_{2},\\ldots,e_{2n}\\}$ of $\\mathbb{R}^{2n}$ , then

\\omega^{n}=n!Pf(A)e_{1}\\wedge e_{2}\\wedge\\cdots\\wedge e_{2n},

where $\\omega^{n}$ denotes exterior product of $n$ copies of $\\omega$ .

Identities

For any antisymmetric $2n\\times 2n$ matrix $A$ ’ and any $2n\\times 2n$ matrix $B$

Pf(A)^{2}=\\det(A)

Pf(BAB^{T})=\\det(B)Pf(A)