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PfaffianDet[m]

gives the Pfaffian determinant of the antisymmetric matrix m.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Method  
Properties & Relations  
See Also
Related Guides
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PfaffianDet

PfaffianDet[m]

gives the Pfaffian determinant of the antisymmetric matrix m.

Details and Options

  • Antisymmetric matrices are also called skew symmetric.
  • The Pfaffian of a antihermitian matrix is defined by , where is SymmetricGroup[2n] and is Signature[α]. »
  • For any antisymmetric matrix , is equal to the determinant of . »
  • The Pfaffian of an odd dimensional matrix is always 0. »
  • The following options can be give
  • Method "ParlettReid"method to use
  • Possible values for the Method option include:
  • "Det"uses the built-in Det
    "Hessenberg"Hessenberg decomposition
    "Householder"Householder tridiagonalization
    "ParlettReid"Parlett-Reid tridiagonalization
    "Pauli"uses the second Pauli matrix
  • 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 matrix is computed in terms of the eigenvalues of TemplateBox[{{(, {{sigma, _, 2}, tensor , {ⅈ, _, n}}, )}}, Transpose].m, where is PauliMatrix[2] and is IdentityMatrix[n].
  • For Method"Det", PfaffianDet is calculated as Sqrt[Det[m]] and is not guaranteed to conform in sign with values obtained by other methods.

Examples

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Basic Examples  (2)

Compute the Pfaffian of an antisymmetric matrix:

Wolfram Language code: PfaffianDet[(| | | | | | -- | -- | -- | - | | 0 | 3 | 2 | 1 | | -3 | 0 | 5 | 6 | | -2 | -5 | 0 | 4 | | -1 | -6 | -4 | 0 |)]

The Pfaffian is the square root of the determinant:

Wolfram Language code: Det[(| | | | | | -- | -- | -- | - | | 0 | 3 | 2 | 1 | | -3 | 0 | 5 | 6 | | -2 | -5 | 0 | 4 | | -1 | -6 | -4 | 0 |)]

The Pfaffian determinant of a complex antisymmetric matrix:

Wolfram Language code: PfaffianDet[(| | | | -- | - | | 0 | I | | -I | 0 |)]

This also equals to square root of the determinant:

Wolfram Language code: Det[(| | | | -- | - | | 0 | I | | -I | 0 |)]

Scope  (5)

The Pfaffian of a real antisymmetric matrix:

Wolfram Language code: Symmetrize[RandomReal[1, {4, 4}], Antisymmetric[{1, 2}]]//MatrixForm
Wolfram Language code: PfaffianDet[%]

A complex antisymmetric matrix:

Wolfram Language code: Symmetrize[RandomComplex[1 + I, {2, 2}], Antisymmetric[{1, 2}]]//MatrixForm

Find the Pfaffian:

Wolfram Language code: PfaffianDet[Normal@%]

An antisymmetric integer matrix:

Wolfram Language code: Symmetrize[2RandomInteger[{-10, 10}, {6, 6}], Antisymmetric[{1, 2}]]//MatrixForm

Compute the Pfaffian:

Wolfram Language code: PfaffianDet[%]

An antisymmetric symbolic matrix:

Wolfram Language code: m = {{0, a}, {-a, 0}}

Compute the Pfaffian:

Wolfram Language code: PfaffianDet[%]

Pfaffian of a large antisymmetric matrix:

Wolfram Language code: SeedRandom[1234]; m = Symmetrize[RandomReal[1, {50, 50}], Antisymmetric[{1, 2}]]; PfaffianDet[m]

Options  (2)

Method  (2)

An antisymmetric matrix:

Wolfram Language code: mat = (# - Transpose[#])&@RandomReal[1, {8, 8}]

Compute the Pfaffian using several methods:

Wolfram Language code: pfafs = Table[PfaffianDet[mat, Method -> method], {method, {"ParlettReid", "Householder", "Hessenberg", "Pauli", "Det"}}]

Check that they agree up to sign:

Wolfram Language code: Abs[Chop[pfafs]]

A complex-valued antisymmetric matrix:

Wolfram Language code: mat = (# - Transpose[#])&@RandomComplex[1 + I, {8, 8}];

Compute the Pfaffian using several methods:

Wolfram Language code: pfafs = Table[PfaffianDet[mat, Method -> method], {method, {"ParlettReid", "Householder", "Pauli", "Det"}}]

Check that they are in numerical agreement:

Wolfram Language code: Max[Abs[#1 - #2] / (Abs[#1] + Abs[#2])&@@@Subsets[pfafs ^ 2, {2}]]

Properties & Relations  (8)

The square of the Pfaffian determinant equals the determinant:

Wolfram Language code: m = Normal@Symmetrize[RandomReal[1, {4, 4}], Antisymmetric[{1, 2}]]; PfaffianDet[m]^2 == Det[m]

This extends to complex-valued matrices:

Wolfram Language code: m = Symmetrize[RandomComplex[1 + I, {4, 4}], Antisymmetric[{1, 2}]]; PfaffianDet[m]^2 == Det[m]

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

Wolfram Language code: PfaffianDet[(| | | | | -- | -- | - | | 0 | a | b | | -a | 0 | c | | -b | -c | 0 |)]

The Pfaffian determinant obeys PfaffianDet[m]==PfaffianDet[Transpose[m]]:

Wolfram Language code: m = Normal@Symmetrize[RandomComplex[1 + I, {4, 4}], Antisymmetric[{1, 2}]];
Wolfram Language code: PfaffianDet[m] == PfaffianDet[m]

If is a antisymmetric matrix and is a real number, then :

Wolfram Language code: m = Normal@Symmetrize[RandomComplex[1 + I, {4, 4}], Antisymmetric[{1, 2}]]; λ = RandomReal[]; PfaffianDet[λ m] == λ ^(Length[m]/2)PfaffianDet[m]

For a real antisymmetric matrix m and an arbitrary real matrix x, x.PfaffianDet[x.m.Transpose[x]]==Det[x]*PfaffianDet[m]:

Wolfram Language code: n = 3; m = Symmetrize[RandomReal[1, {2n, 2n}, WorkingPrecision -> 10], Antisymmetric[{1, 2}]]; x = RandomReal[1, {2n, 2n}, WorkingPrecision -> 10]; PfaffianDet[ x.m.Transpose[x]] == Det[x]PfaffianDet[m]

If is , PfaffianDet[m] can be computed as using Permutations and Signature:

Wolfram Language code: pfaf[m_] := With[{n = Length[m] / 2}, Sum[ (1/2^n n!)Signature[s]Product[m[[s[[2i - 1]], s[[2i]]]], {i, n}], {s, Permutations[Range[2n]]}]]

Confirm for a generic symbolic matrix:

Wolfram Language code: m = Symmetrize[Table[a[i, j], {i, 6}, {j, 6}], Antisymmetric[{1, 2}]]; PfaffianDet[m] == pfaf[m]//Simplify

If is , PfaffianDet[m] can be computed as , where is LeviCivitaTensor[2n], using TensorProduct and TensorContract:

Wolfram Language code: m = Symmetrize[Table[a[i, j], {i, 4}, {j, 4}], Antisymmetric[{1, 2}]]; PfaffianDet[m] == (1/2^22!)TensorContract[mmLeviCivitaTensor[4], {{1, 5}, {2, 6}, {3, 7}, {4, 8}}]//Simplify

Generalize to higher dimensions, using Inactive[TensorProduct] for efficiency:

Wolfram Language code: pfaf[m_] := With[{n = Length[m] / 2}, (1/2^nn!)Activate[TensorContract[Inactive[TensorProduct]@@Append[ConstantArray[m, n], LeviCivitaTensor[2n]], Table[{k, k + 2n}, {k, 2n}]]]]
Wolfram Language code: m = Symmetrize[Table[a[i, j], {i, 6}, {j, 6}], Antisymmetric[{1, 2}]]; PfaffianDet[m] == pfaf[m]//Simplify

For an antisymmetric tridiagonal matrix, the Pfaffian equals the product of the superdiagonal matrix:

Wolfram Language code: AntisymmetricTridiagonalMatrix[symb_, n_] := 2Symmetrize[SparseArray[{{i_, j_} /; j - i == 1 && Mod[j, 2] == 0 -> Subscript[symb, j / 2]}, {2n, 2n}], Antisymmetric[{1, 2}]]; MatrixForm[mat = AntisymmetricTridiagonalMatrix[a, 3]]
Wolfram Language code: PfaffianDet[mat]
Wolfram Research (2026), PfaffianDet, Wolfram Language function, https://reference.wolfram.com/language/ref/PfaffianDet.html.

Text

Wolfram Research (2026), PfaffianDet, Wolfram Language function, https://reference.wolfram.com/language/ref/PfaffianDet.html.

CMS

Wolfram Language. 2026. "PfaffianDet." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PfaffianDet.html.

APA

Wolfram Language. (2026). PfaffianDet. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PfaffianDet.html

BibTeX

@misc{reference.wolfram_2026_pfaffiandet, author="Wolfram Research", title="{PfaffianDet}", year="2026", howpublished="\url{https://reference.wolfram.com/language/ref/PfaffianDet.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_pfaffiandet, organization={Wolfram Research}, title={PfaffianDet}, year={2026}, url={https://reference.wolfram.com/language/ref/PfaffianDet.html}, note=[Accessed: 12-June-2026]}