Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
ALGORITHM:
Use PARI whenever the method self._adjoint is included to do so in an inheriting class. Otherwise,
use a generic division-free algorithm to compute the characteristic polynomial and hence the adjoint.
The result is cached.
EXAMPLES:
sage: M = Matrix(ZZ,2,2,[5,2,3,4]) ; M
[5 2]
[3 4]
sage: N = M.adjoint() ; N
[ 4 -2]
[-3 5]
sage: M * N
[14 0]
[ 0 14]
sage: N * M
[14 0]
[ 0 14]
sage: M = Matrix(QQ,2,2,[5/3,2/56,33/13,41/10]) ; M
[ 5/3 1/28]
[33/13 41/10]
sage: N = M.adjoint() ; N
[ 41/10 -1/28]
[-33/13 5/3]
sage: M * N
[7363/1092 0]
[ 0 7363/1092]
AUTHORS:
•Unknown: No author specified in the file from 2009-06-25
•Sebastian Pancratz (2009-06-25): Reflecting the change that _adjoint is now implemented in this
class
as_sum_of_permutations()
Returns the current matrix as a sum of permutation matrices
According to the Birkhoff-von Neumann Theorem, any bistochastic matrix can be written as a positive
sum of permutation matrices, which also means that the polytope of bistochastic matrices is integer.
As a non-bistochastic matrix can obviously not be written as a sum of permutations, this theorem is an
equivalence.
This function, given a bistochastic matrix, returns the corresponding decomposition.
•bistochastic_as_sum_of_permutations – for more information on this method.
EXAMPLE:
We create a bistochastic matrix from a convex sum of permutations, then try to deduce the decomposition
from the matrix
sage: L = []
sage: L.append((9,Permutation([4, 1, 3, 5, 2])))
sage: L.append((6,Permutation([5, 3, 4, 1, 2])))
sage: L.append((3,Permutation([3, 1, 4, 2, 5])))
sage: L.append((2,Permutation([1, 4, 2, 3, 5])))
sage: M = sum([c * p.to_matrix() for (c,p) in L])
sage: decomp = sage.combinat.permutation.bistochastic_as_sum_of_permutations(M)
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