from sympy.core import S
from sympy.core.sympify import _sympify
from sympy.functions import KroneckerDelta
from .matexpr import MatrixExpr
from .special import ZeroMatrix, Identity, OneMatrix
class PermutationMatrix(MatrixExpr):
"""A Permutation Matrix
Parameters
==========
perm : Permutation
The permutation the matrix uses.
The size of the permutation determines the matrix size.
See the documentation of
:class:`sympy.combinatorics.permutations.Permutation` for
the further information of how to create a permutation object.
Examples
========
>>> from sympy import Matrix, PermutationMatrix
>>> from sympy.combinatorics import Permutation
Creating a permutation matrix:
>>> p = Permutation(1, 2, 0)
>>> P = PermutationMatrix(p)
>>> P = P.as_explicit()
>>> P
Matrix([
[0, 1, 0],
[0, 0, 1],
[1, 0, 0]])
Permuting a matrix row and column:
>>> M = Matrix([0, 1, 2])
>>> Matrix(P*M)
Matrix([
[1],
[2],
[0]])
>>> Matrix(M.T*P)
Matrix([[2, 0, 1]])
See Also
========
sympy.combinatorics.permutations.Permutation
"""
def __new__(cls, perm):
from sympy.combinatorics.permutations import Permutation
perm = _sympify(perm)
if not isinstance(perm, Permutation):
raise ValueError(
"{} must be a SymPy Permutation instance.".format(perm))
return super().__new__(cls, perm)
@property
def shape(self):
size = self.args[0].size
return (size, size)
@property
def is_Identity(self):
return self.args[0].is_Identity
def doit(self, **hints):
if self.is_Identity:
return Identity(self.rows)
return self
def _entry(self, i, j, **kwargs):
perm = self.args[0]
return KroneckerDelta(perm.apply(i), j)
def _eval_power(self, exp):
return PermutationMatrix(self.args[0] ** exp).doit()
def _eval_inverse(self):
return PermutationMatrix(self.args[0] ** -1)
_eval_transpose = _eval_adjoint = _eval_inverse
def _eval_determinant(self):
sign = self.args[0].signature()
if sign == 1:
return S.One
elif sign == -1:
return S.NegativeOne
raise NotImplementedError
def _eval_rewrite_as_BlockDiagMatrix(self, *args, **kwargs):
from sympy.combinatorics.permutations import Permutation
from .blockmatrix import BlockDiagMatrix
perm = self.args[0]
full_cyclic_form = perm.full_cyclic_form
cycles_picks = []
# Stage 1. Decompose the cycles into the blockable form.
a, b, c = 0, 0, 0
flag = False
for cycle in full_cyclic_form:
l = len(cycle)
m = max(cycle)
if not flag:
if m + 1 > a + l:
flag = True
temp = [cycle]
b = m
c = l
else:
cycles_picks.append([cycle])
a += l
else:
if m > b:
if m + 1 == a + c + l:
temp.append(cycle)
cycles_picks.append(temp)
flag = False
a = m+1
else:
b = m
temp.append(cycle)
c += l
else:
if b + 1 == a + c + l:
temp.append(cycle)
cycles_picks.append(temp)
flag = False
a = b+1
else:
temp.append(cycle)
c += l
# Stage 2. Normalize each decomposed cycles and build matrix.
p = 0
args = []
for pick in cycles_picks:
new_cycles = []
l = 0
for cycle in pick:
new_cycle = [i - p for i in cycle]
new_cycles.append(new_cycle)
l += len(cycle)
p += l
perm = Permutation(new_cycles)
mat = PermutationMatrix(perm)
args.append(mat)
return BlockDiagMatrix(*args)
class MatrixPermute(MatrixExpr):
r"""Symbolic representation for permuting matrix rows or columns.
Parameters
==========
perm : Permutation, PermutationMatrix
The permutation to use for permuting the matrix.
The permutation can be resized to the suitable one,
axis : 0 or 1
The axis to permute alongside.
If `0`, it will permute the matrix rows.
If `1`, it will permute the matrix columns.
Notes
=====
This follows the same notation used in
:meth:`sympy.matrices.common.MatrixCommon.permute`.
Examples
========
>>> from sympy import Matrix, MatrixPermute
>>> from sympy.combinatorics import Permutation
Permuting the matrix rows:
>>> p = Permutation(1, 2, 0)
>>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> B = MatrixPermute(A, p, axis=0)
>>> B.as_explicit()
Matrix([
[4, 5, 6],
[7, 8, 9],
[1, 2, 3]])
Permuting the matrix columns:
>>> B = MatrixPermute(A, p, axis=1)
>>> B.as_explicit()
Matrix([
[2, 3, 1],
[5, 6, 4],
[8, 9, 7]])
See Also
========
sympy.matrices.common.MatrixCommon.permute
"""
def __new__(cls, mat, perm, axis=S.Zero):
from sympy.combinatorics.permutations import Permutation
mat = _sympify(mat)
if not mat.is_Matrix:
raise ValueError(
"{} must be a SymPy matrix instance.".format(perm))
perm = _sympify(perm)
if isinstance(perm, PermutationMatrix):
perm = perm.args[0]
if not isinstance(perm, Permutation):
raise ValueError(
"{} must be a SymPy Permutation or a PermutationMatrix " \
"instance".format(perm))
axis = _sympify(axis)
if axis not in (0, 1):
raise ValueError("The axis must be 0 or 1.")
mat_size = mat.shape[axis]
if mat_size != perm.size:
try:
perm = perm.resize(mat_size)
except ValueError:
raise ValueError(
"Size does not match between the permutation {} "
"and the matrix {} threaded over the axis {} "
"and cannot be converted."
.format(perm, mat, axis))
return super().__new__(cls, mat, perm, axis)
def doit(self, deep=True, **hints):
mat, perm, axis = self.args
if deep:
mat = mat.doit(deep=deep, **hints)
perm = perm.doit(deep=deep, **hints)
if perm.is_Identity:
return mat
if mat.is_Identity:
if axis is S.Zero:
return PermutationMatrix(perm)
elif axis is S.One:
return PermutationMatrix(perm**-1)
if isinstance(mat, (ZeroMatrix, OneMatrix)):
return mat
if isinstance(mat, MatrixPermute) and mat.args[2] == axis:
return MatrixPermute(mat.args[0], perm * mat.args[1], axis)
return self
@property
def shape(self):
return self.args[0].shape
def _entry(self, i, j, **kwargs):
mat, perm, axis = self.args
if axis == 0:
return mat[perm.apply(i), j]
elif axis == 1:
return mat[i, perm.apply(j)]
def _eval_rewrite_as_MatMul(self, *args, **kwargs):
from .matmul import MatMul
mat, perm, axis = self.args
deep = kwargs.get("deep", True)
if deep:
mat = mat.rewrite(MatMul)
if axis == 0:
return MatMul(PermutationMatrix(perm), mat)
elif axis == 1:
return MatMul(mat, PermutationMatrix(perm**-1))