def test_connected_components():
a, b, c, d, e, f, g, h, i, j, k, l, m = symbols('a:m')
M = Matrix([[a, 0, 0, 0, b, 0, 0, 0, 0, 0, c, 0, 0],
[0, d, 0, 0, 0, e, 0, 0, 0, 0, 0, f, 0],
[0, 0, g, 0, 0, 0, h, 0, 0, 0, 0, 0, i],
[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[m, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[0, m, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
[0, 0, m, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
[j, 0, 0, 0, k, 0, 0, 1, 0, 0, l, 0, 0],
[0, j, 0, 0, 0, k, 0, 0, 1, 0, 0, l, 0],
[0, 0, j, 0, 0, 0, k, 0, 0, 1, 0, 0, l],
[0, 0, 0, 0, d, 0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, d, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, d, 0, 0, 0, 0, 0, 1]])
cc = M.connected_components()
assert cc == [[0, 4, 7, 10], [1, 5, 8, 11], [2, 6, 9, 12], [3]]
P, B = M.connected_components_decomposition()
p = Permutation([0, 4, 7, 10, 1, 5, 8, 11, 2, 6, 9, 12, 3])
assert P == PermutationMatrix(p)
B0 = Matrix([[a, b, 0, c], [m, 1, 0, 0], [j, k, 1, l], [0, d, 0, 1]])
B1 = Matrix([[d, e, 0, f], [m, 1, 0, 0], [j, k, 1, l], [0, d, 0, 1]])
B2 = Matrix([[g, h, 0, i], [m, 1, 0, 0], [j, k, 1, l], [0, d, 0, 1]])
B3 = Matrix([[1]])
assert B == BlockDiagMatrix(B0, B1, B2, B3)
def test_PermutationMatrix_rewrite_BlockDiagMatrix():
P = PermutationMatrix(Permutation([0, 1, 2, 3, 4, 5]))
P0 = PermutationMatrix(Permutation([0]))
assert P.rewrite(BlockDiagMatrix) == \
BlockDiagMatrix(P0, P0, P0, P0, P0, P0)
P = PermutationMatrix(Permutation([0, 1, 3, 2, 4, 5]))
P10 = PermutationMatrix(Permutation(0, 1))
assert P.rewrite(BlockDiagMatrix) == \
BlockDiagMatrix(P0, P0, P10, P0, P0)
P = PermutationMatrix(Permutation([1, 0, 3, 2, 5, 4]))
assert P.rewrite(BlockDiagMatrix) == \
BlockDiagMatrix(P10, P10, P10)
P = PermutationMatrix(Permutation([0, 4, 3, 2, 1, 5]))
P3210 = PermutationMatrix(Permutation([3, 2, 1, 0]))
assert P.rewrite(BlockDiagMatrix) == \
BlockDiagMatrix(P0, P3210, P0)
P = PermutationMatrix(Permutation([0, 4, 2, 3, 1, 5]))
P3120 = PermutationMatrix(Permutation([3, 1, 2, 0]))
assert P.rewrite(BlockDiagMatrix) == \
BlockDiagMatrix(P0, P3120, P0)
P = PermutationMatrix(Permutation(0, 3)(1, 4)(2, 5))
assert P.rewrite(BlockDiagMatrix) == BlockDiagMatrix(P)
def test_invertible_BlockDiagMatrix():
assert ask(Q.invertible(BlockDiagMatrix(Identity(3), Identity(5)))) == True
assert ask(Q.invertible(BlockDiagMatrix(ZeroMatrix(3, 3),
Identity(5)))) == False
assert ask(Q.invertible(BlockDiagMatrix(Identity(3),
OneMatrix(5, 5)))) == False