Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
and j, we denote the (i, j)-th entry of A by a i,j . Let M denote the set of all perfect matchings of the
set {1, 2, . . . , 2n} (see sage.combinat.perfect_matching.PerfectMatchings ). For every
matching m ∈ M, define the sign sign(m) of m by writing m as {{i 1 , j 1 }, {i 2 , j 2 }, . . . , {i n , j n }} with
i k < j k for all k, and setting sign(m) to be the sign of the permutation (i 1 , j 1 , i 2 , j 2 , . . . , i n , j n ) (written
here in one-line notation). For every matching m ∈ M, define the weight w(m) of m by writing m as
{{i 1 , j 1 }, {i 2 , j 2 }, . . . , {i n , j n }} with i k < j k for all k, and setting w(m) = a i1,j 1
a i2,j 2
· · · a in,j n
. Now,
the Pfaffian of the matrix A is defined to be the sum
∑
sign(m)w(m).
m∈M
The Pfaffian of A is commonly denoted by Pf(A). It is well-known that (Pf(A)) 2 = det A for every
alternating matrix A, and that Pf(U T AU) = det U · Pf(A) for any n × n matrix U and any alternating
n × n matrix A.
See [Kn95], [DW95] and [Rote2001], just to name three sources, for further properties of Pfaffians.
ALGORITHM:
The current implementation uses the definition given above. It checks alternatingness of the matrix self
only if check is True (this is important because even if self is alternating, a non-discrete base ring
might prevent Sage from being able to check this).
REFERENCES:
Todo
Implement faster algorithms, including a division-free one. Does [Rote2001], section 3.3 give one
Check the implementation of the matchings used here for performance
EXAMPLES:
A 3 × 3 alternating matrix has Pfaffian 0 independently of its entries:
sage: MSp = MatrixSpace(Integers(27), 3)
sage: A = MSp([0, 2, -3, -2, 0, 8, 3, -8, 0])
sage: A.pfaffian()
0
sage: parent(A.pfaffian())
Ring of integers modulo 27
The Pfaffian of a 2 × 2 alternating matrix is just its northeast entry:
sage: MSp = MatrixSpace(QQ, 2)
sage: A = MSp([0, 4, -4, 0])
sage: A.pfaffian()
4
sage: parent(A.pfaffian())
Rational Field
The Pfaffian of a 0 × 0 alternating matrix is 1:
sage: MSp = MatrixSpace(ZZ, 0)
sage: A = MSp([])
sage: A.pfaffian()
1
sage: parent(A.pfaffian())
Integer Ring
Let us compute the Pfaffian of a generic 4 × 4 alternating matrix:
221