Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
•self – matrix of size m × n with m ≤ n
OUTPUT:
The permanental k-minor of the matrix self.
EXAMPLES:
sage: M = MatrixSpace(ZZ,4,4)
sage: A = M([1,0,1,0,1,0,1,0,1,0,10,10,1,0,1,1])
sage: A.permanental_minor(2)
114
sage: M = MatrixSpace(ZZ,3,6)
sage: A = M([1,1,1,1,0,0,0,1,1,1,1,0,0,0,1,1,1,1])
sage: A.permanental_minor(0)
1
sage: A.permanental_minor(1)
12
sage: A.permanental_minor(2)
40
sage: A.permanental_minor(3)
36
Note that if k = m = n, the permanental k-minor equals per(A):
sage: A.permanent()
36
sage: A.permanental_minor(5)
0
For C the “complement” of A:
sage: M = MatrixSpace(ZZ,3,6)
sage: C = M([0,0,0,0,1,1,1,0,0,0,0,1,1,1,0,0,0,0])
sage: m, n = 3, 6
sage: sum([(-1)^k * C.permanental_minor(k)*factorial(n-k)/factorial(n-m) for k in range(m+1)
36
See Theorem 7.2.1 of Brualdi and Ryser: Combinatorial Matrix Theory: per(A)
AUTHORS:
•Jaap Spies (2006-02-19)
pfaffian(algorithm=None, check=True)
Return the Pfaffian of self, assuming that self is an alternating matrix.
INPUT:
•algorithm – string, the algorithm to use; currently the following algorithms have been implemented:
–’definition’ - using the definition given by perfect matchings
•check (default: True) – Boolean determining whether to check self for alternatingness and
squareness. This has to be set to False if self is defined over a non-discrete ring.
The Pfaffian of an alternating matrix is defined as follows:
Let A be an alternating k × k matrix over a commutative ring. (Here, “alternating” means that A T = −A
and that the diagonal entries of A are zero.) If k is odd, then the Pfaffian of the matrix A is defined to
be 0. Let us now define it when k is even. In this case, set n = k/2 (this is an integer). For every i
220 Chapter 7. Base class for matrices, part 2