Function Repository Resource:

Pfaffian

Compute the Pfaffian of an antisymmetric (skew-symmetric) matrix

Contributed by: Wolfram Staff (original content by M. Wimmer)

ResourceFunction["Pfaffian"][m]

gives the Pfaffian of a skew-symmetric matrix m.

Details and Options

The Pfaffian of a skew-symmetric matrix m is an integer-coefficient polynomial in the entries of m whose square is the determinant of m.

Skew-symmetric matrices are also called antisymmetric.

ResourceFunction["Pfaffian"] takes the Method option with the following possible values:

"ParlettReid"

Parlett–Reid tridiagonalization

"Householder"

Householder tridiagonalization

"Hessenberg"

Hessenberg decomposition

"Pauli"

uses the second Pauli matrix

"Det"

uses the built-in Det

The "Householder" and "Hessenberg" methods are applicable only to explicitly-numeric matrices m.

For Method→"Hessenberg", the matrix m must be real.

With Method→"Pauli", the Pfaffian of a 2n⨯2n matrix m is computed in terms of the eigenvalues of

, where σ₂ is the second Pauli matrix and I_n is the n×n identity matrix.

For Method→"Det", ResourceFunction["Pfaffian"] is calculated as Sqrt[Det[m]] and is not guaranteed to conform in sign with values obtained by other methods.

Examples

Basic Examples (1)

Pfaffian of an antisymmetric matrix:

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ResourceFunction["Pfaffian"][( {
{0, a, b, c},
{-a, 0, d, e},
{-b, -d, 0, f},
{-c, -e, -f, 0}
} )] // Simplify

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Scope (3)

The Pfaffian of a real antisymmetric matrix:

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A complex matrix:

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A symbolic matrix:

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ResourceFunction["Pfaffian"][( {
{0, a, b, c, d, e},
{-a, 0, f, g, h, i},
{-b, -f, 0, j, k, l},
{-c, -g, -j, 0, m, n},
{-d, -h, -k, -m, 0, o},
{-e, -i, -l, -n, -o, 0}
} )] // Simplify

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Options (3)

Method (3)

Methods applicable to real matrices:

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Methods applicable to complex matrices:

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$Max[Abs[#1 - #2]/(Abs[#1] + Abs[#2]) & @@@ Subsets[%^2, {2}]]$

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Methods applicable to symbolic matrices:

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mat = ( {
{0, a, b, c, d, e},
{-a, 0, f, g, h, i},
{-b, -f, 0, j, k, l},
{-c, -g, -j, 0, m, n},
{-d, -h, -k, -m, 0, o},
{-e, -i, -l, -n, -o, 0}
} )

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Properties and Relations (6)

The Pfaffian of an antisymmetric matrix of odd dimension is zero:

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For a 2n×2n skew-symmetric matrix m, the Pfaffian of the matrix transpose m^T is equal to (-1)ⁿPfaffian[m]:

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$AntisymmetricMatrix[symb_, n_] := # - Transpose[#] &@ SparseArray[{{i_, j_} /; j > i -> Subscript[symb, i, j]}, {2 n, 2 n}]$

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$SameQ @@ Simplify[{ResourceFunction["Pfaffian"][Transpose[mat]], (-1)^ n ResourceFunction["Pfaffian"][mat]}]$

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For such matrices, the Pfaffian of λm is equal to λⁿPfaffian[m]:

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$SameQ @@ Simplify[{ResourceFunction["Pfaffian"][\[Lambda] mat], \[Lambda]^ n ResourceFunction["Pfaffian"][mat]}]$

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Also, for such matrices, the square of the Pfaffian is equal to the determinant of the matrix:

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$SameQ @@ Simplify[{ResourceFunction["Pfaffian"][ mat]^2, Det[mat]}]$

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For a 2n×2n skew-symmetric matrix m and an arbitrary 2n×2n matrix x, Pfaffian[x.m.x^T]⩵Det[x]*Pfaffian[m]:

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The Pfaffian of a 2n×2n skew-symmetric block diagonal matrix is the product of its non-zero superdiagonal values:

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$AntisymmetricBlockDiagonalMatrix[symb_, n_] := # - Transpose[#] &@ SparseArray[{{i_, j_} /; j - i == 1 && EvenQ[j] -> Subscript[symb, j/2]}, {2 n, 2 n}]$

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The Pfaffian of a 2n×2n skew-symmetric tridiagonal matrix is the product of the entries at odd positions on the superdiagonal:

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$AntisymmetricTridiagonalMatrix[symb_, n_] := # - Transpose[#] &@ SparseArray[{{i_, j_} /; j - i == 1 -> Subscript[symb, j - 1]}, {2 n, 2 n}]$

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Reducing an antisymmetric matrix to its tridiagonal form gives a numerically stable way to compute the Pfaffian from the superdiagonal values:

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Tridiagonalize using the resource function SkewLTLDecomposition:

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Using the resource function SkewTridiagonalDecomposition:

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Version History

1.0.0 – 06 November 2020

Source Metadata

Citation:
- Wimmer, M., "Algorithm 923: Efficient Numerical Computation of the Pfaffian for Dense and Banded Skew-Symmetric Matrices." ACM Trans. Math. Softw., vol. 38, no. 4, 1–17, 2012. DOI: 10.1145/2331130.2331138.

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License Information

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