pfaffian -- Pfaffian of a skew-symmetric matrix

Usage:

pfaffian M
Inputs:
- M, a matrix, must be skew-symmetric;
Outputs:
- a number or a ring element, the Pfaffian of M;

Description

The Pfaffian of a $2n\times 2n$ skew-symmetric matrix $M=(m_{ij})$ is $$\frac{1}{2^nn!}\sum_{\sigma\in S_{2n}}\operatorname{sgn}(\sigma)\prod_{i=1}^nm_{\sigma(2i-1),\sigma(2i)},$$ where $S_n$ is the symmetric group on $2n$ elements.

i1 : M = matrix {{0, 1, 2, 3}, {-1, 0, 4, 5}, {-2, -4, 0, 6}, {-3, -5, -6, 0}}

o1 = | 0  1  2  3 |
     | -1 0  4  5 |
     | -2 -4 0  6 |
     | -3 -5 -6 0 |

              4       4
o1 : Matrix ZZ  <-- ZZ

i2 : pfaffian M

o2 = 8

The square of the Pfaffian is the determinant.

i3 : determinant M

o3 = 64

Skew-symmetric matrices with an odd number of rows and columns have Pfaffian zero.

i4 : M = matrix {{0, 1, 2}, {-1, 0, 3}, {-2, -3, 0}}

o4 = | 0  1  2 |
     | -1 0  3 |
     | -2 -3 0 |

              3       3
o4 : Matrix ZZ  <-- ZZ

i5 : pfaffian M

o5 = 0

Ways to use pfaffian:

pfaffian(Matrix)

For the programmer

The object pfaffian is a method function.

The source of this document is in Macaulay2Doc/functions/pfaffian-doc.m2:36:0.

pfaffian -- Pfaffian of a skew-symmetric matrix

Description

See also

Ways to use pfaffian:

For the programmer