def test_squareBlockMatrix():
A = MatrixSymbol("A", n, n)
B = MatrixSymbol("B", n, m)
C = MatrixSymbol("C", m, n)
D = MatrixSymbol("D", m, m)
X = BlockMatrix([[A, B], [C, D]])
Y = BlockMatrix([[A]])
assert X.is_square
assert block_collapse(X + Identity(m + n)) == BlockMatrix([[A + Identity(n), B], [C, D + Identity(m)]])
Q = X + Identity(m + n)
assert (X + MatrixSymbol("Q", n + m, n + m)).is_MatAdd
assert (X * MatrixSymbol("Q", n + m, n + m)).is_MatMul
assert block_collapse(Y.I) == A.I
assert block_collapse(X.inverse()) == BlockMatrix(
[[(-B * D.I * C + A).I, -A.I * B * (D + -C * A.I * B).I], [-(D - C * A.I * B).I * C * A.I, (D - C * A.I * B).I]]
)
assert isinstance(X.inverse(), Inverse)
assert not X.is_Identity
Z = BlockMatrix([[Identity(n), B], [C, D]])
assert not Z.is_Identity
def test_BlockDiagMatrix():
A = MatrixSymbol("A", n, n)
B = MatrixSymbol("B", m, m)
C = MatrixSymbol("C", l, l)
M = MatrixSymbol("M", n + m + l, n + m + l)
X = BlockDiagMatrix(A, B, C)
Y = BlockDiagMatrix(A, 2 * B, 3 * C)
assert X.blocks[1, 1] == B
assert X.shape == (n + m + l, n + m + l)
assert all(
X.blocks[i, j].is_ZeroMatrix if i != j else X.blocks[i, j] in [A, B, C] for i in range(3) for j in range(3)
)
assert X.__class__(*X.args) == X
assert isinstance(block_collapse(X.I * X), Identity)
assert bc_matmul(X * X) == BlockDiagMatrix(A * A, B * B, C * C)
assert block_collapse(X * X) == BlockDiagMatrix(A * A, B * B, C * C)
# XXX: should be == ??
assert block_collapse(X + X).equals(BlockDiagMatrix(2 * A, 2 * B, 2 * C))
assert block_collapse(X * Y) == BlockDiagMatrix(A * A, 2 * B * B, 3 * C * C)
assert block_collapse(X + Y) == BlockDiagMatrix(2 * A, 3 * B, 4 * C)
# Ensure that BlockDiagMatrices can still interact with normal MatrixExprs
assert (X * (2 * M)).is_MatMul
assert (X + (2 * M)).is_MatAdd
assert (X._blockmul(M)).is_MatMul
assert (X._blockadd(M)).is_MatAdd
def test_squareBlockMatrix():
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, n)
D = MatrixSymbol('D', m, m)
X = BlockMatrix([[A, B], [C, D]])
Y = BlockMatrix([[A]])
assert X.is_square
assert (block_collapse(X + Identity(m + n)) == BlockMatrix(
[[A + Identity(n), B], [C, D + Identity(m)]]))
Q = X + Identity(m + n)
assert (X + MatrixSymbol('Q', n + m, n + m)).is_MatAdd
assert (X * MatrixSymbol('Q', n + m, n + m)).is_MatMul
assert block_collapse(Y.I) == A.I
assert block_collapse(X.inverse()) == BlockMatrix([[
(-B * D.I * C + A).I, -A.I * B * (D + -C * A.I * B).I
], [-(D - C * A.I * B).I * C * A.I, (D - C * A.I * B).I]])
assert isinstance(X.inverse(), Inverse)
assert not X.is_Identity
Z = BlockMatrix([[Identity(n), B], [C, D]])
assert not Z.is_Identity
def test_BlockMatrix_inverse():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, n)
C = MatrixSymbol('C', m, m)
D = MatrixSymbol('D', m, n)
X = BlockMatrix([[A, B], [C, D]])
assert X.is_square
assert isinstance(block_collapse(X.inverse()),
Inverse) # Can't inverse when A, D aren't square
# test code path for non-invertible D matrix
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, n)
D = OneMatrix(m, m)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[
A.I + A.I * B * (D - C * A.I * B).I * C * A.I,
-A.I * B * (D - C * A.I * B).I
],
[-(D - C * A.I * B).I * C * A.I, (D - C * A.I * B).I],
])
# test code path for non-invertible A matrix
A = OneMatrix(n, n)
D = MatrixSymbol('D', m, m)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[(A - B * D.I * C).I, -(A - B * D.I * C).I * B * D.I],
[
-D.I * C * (A - B * D.I * C).I,
D.I + D.I * C * (A - B * D.I * C).I * B * D.I
],
])
def test_squareBlockMatrix():
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, n)
D = MatrixSymbol('D', m, m)
X = BlockMatrix([[A, B], [C, D]])
Y = BlockMatrix([[A]])
assert X.is_square
assert (block_collapse(X + Identity(m + n)) ==
BlockMatrix([[A + Identity(n), B], [C, D + Identity(m)]]))
Q = X + Identity(m + n)
assert block_collapse(Q.inverse()) == Inverse(block_collapse(Q))
assert (X + MatrixSymbol('Q', n + m, n + m)).is_MatAdd
assert (X * MatrixSymbol('Q', n + m, n + m)).is_MatMul
assert Y.I.blocks[0, 0] == A.I
assert X.inverse(expand=True) == BlockMatrix([
[(-B*D.I*C + A).I, -A.I*B*(D + -C*A.I*B).I],
[-(D - C*A.I*B).I*C*A.I, (D - C*A.I*B).I]])
assert isinstance(X.inverse(expand=False), Inverse)
assert isinstance(X.inverse(), Inverse)
assert not X.is_Identity
Z = BlockMatrix([[Identity(n), B], [C, D]])
assert not Z.is_Identity
def test_BlockDiagMatrix():
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', m, m)
C = MatrixSymbol('C', l, l)
M = MatrixSymbol('M', n + m + l, n + m + l)
X = BlockDiagMatrix(A, B, C)
Y = BlockDiagMatrix(A, 2 * B, 3 * C)
assert X.blocks[1, 1] == B
assert X.shape == (n + m + l, n + m + l)
assert all(
X.blocks[i, j].is_ZeroMatrix if i != j else X.blocks[i,
j] in [A, B, C]
for i in range(3) for j in range(3))
assert X.__class__(*X.args) == X
assert isinstance(block_collapse(X.I * X), Identity)
assert bc_matmul(X * X) == BlockDiagMatrix(A * A, B * B, C * C)
assert block_collapse(X * X) == BlockDiagMatrix(A * A, B * B, C * C)
#XXX: should be == ??
assert block_collapse(X + X).equals(BlockDiagMatrix(2 * A, 2 * B, 2 * C))
assert block_collapse(X * Y) == BlockDiagMatrix(A * A, 2 * B * B,
3 * C * C)
assert block_collapse(X + Y) == BlockDiagMatrix(2 * A, 3 * B, 4 * C)
# Ensure that BlockDiagMatrices can still interact with normal MatrixExprs
assert (X * (2 * M)).is_MatMul
assert (X + (2 * M)).is_MatAdd
assert (X._blockmul(M)).is_MatMul
assert (X._blockadd(M)).is_MatAdd
def test_block_collapse_type():
bm1 = BlockDiagMatrix(ImmutableMatrix([1]), ImmutableMatrix([2]))
bm2 = BlockDiagMatrix(ImmutableMatrix([3]), ImmutableMatrix([4]))
assert bm1.T.__class__ == BlockDiagMatrix
assert block_collapse(bm1 - bm2).__class__ == BlockDiagMatrix
assert block_collapse(Inverse(bm1)).__class__ == BlockDiagMatrix
assert block_collapse(Transpose(bm1)).__class__ == BlockDiagMatrix
assert bc_transpose(Transpose(bm1)).__class__ == BlockDiagMatrix
assert bc_inverse(Inverse(bm1)).__class__ == BlockDiagMatrix
def test_BlockMatrix_2x2_inverse_symbolic():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, k - m)
C = MatrixSymbol('C', k - n, m)
D = MatrixSymbol('D', k - n, k - m)
X = BlockMatrix([[A, B], [C, D]])
assert X.is_square and X.shape == (k, k)
assert isinstance(block_collapse(
X.I), Inverse) # Can't invert when none of the blocks is square
# test code path where only A is invertible
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, n)
D = ZeroMatrix(m, m)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[A.I + A.I * B * X.schur('A').I * C * A.I, -A.I * B * X.schur('A').I],
[-X.schur('A').I * C * A.I, X.schur('A').I],
])
# test code path where only B is invertible
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, n)
C = ZeroMatrix(m, m)
D = MatrixSymbol('D', m, n)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[-X.schur('B').I * D * B.I, X.schur('B').I],
[B.I + B.I * A * X.schur('B').I * D * B.I, -B.I * A * X.schur('B').I],
])
# test code path where only C is invertible
A = MatrixSymbol('A', n, m)
B = ZeroMatrix(n, n)
C = MatrixSymbol('C', m, m)
D = MatrixSymbol('D', m, n)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[-C.I * D * X.schur('C').I, C.I + C.I * D * X.schur('C').I * A * C.I],
[X.schur('C').I, -X.schur('C').I * A * C.I],
])
# test code path where only D is invertible
A = ZeroMatrix(n, n)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, n)
D = MatrixSymbol('D', m, m)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[X.schur('D').I, -X.schur('D').I * B * D.I],
[-D.I * C * X.schur('D').I, D.I + D.I * C * X.schur('D').I * B * D.I],
])
def test_BlockMatrix():
A = MatrixSymbol("A", n, m)
B = MatrixSymbol("B", n, k)
C = MatrixSymbol("C", l, m)
D = MatrixSymbol("D", l, k)
M = MatrixSymbol("M", m + k, p)
N = MatrixSymbol("N", l + n, k + m)
X = BlockMatrix(Matrix([[A, B], [C, D]]))
assert X.__class__(*X.args) == X
# block_collapse does nothing on normal inputs
E = MatrixSymbol("E", n, m)
assert block_collapse(A + 2 * E) == A + 2 * E
F = MatrixSymbol("F", m, m)
assert block_collapse(E.T * A * F) == E.T * A * F
assert X.shape == (l + n, k + m)
assert X.blockshape == (2, 2)
assert transpose(X) == BlockMatrix(Matrix([[A.T, C.T], [B.T, D.T]]))
assert transpose(X).shape == X.shape[::-1]
# Test that BlockMatrices and MatrixSymbols can still mix
assert (X * M).is_MatMul
assert X._blockmul(M).is_MatMul
assert (X * M).shape == (n + l, p)
assert (X + N).is_MatAdd
assert X._blockadd(N).is_MatAdd
assert (X + N).shape == X.shape
E = MatrixSymbol("E", m, 1)
F = MatrixSymbol("F", k, 1)
Y = BlockMatrix(Matrix([[E], [F]]))
assert (X * Y).shape == (l + n, 1)
assert block_collapse(X * Y).blocks[0, 0] == A * E + B * F
assert block_collapse(X * Y).blocks[1, 0] == C * E + D * F
# block_collapse passes down into container objects, transposes, and inverse
assert block_collapse(transpose(X * Y)) == transpose(block_collapse(X * Y))
assert block_collapse(Tuple(X * Y, 2 * X)) == (
block_collapse(X * Y),
block_collapse(2 * X),
)
# Make sure that MatrixSymbols will enter 1x1 BlockMatrix if it simplifies
Ab = BlockMatrix([[A]])
Z = MatrixSymbol("Z", *A.shape)
assert block_collapse(Ab + Z) == A + Z
def test_BlockDiagMatrix_nonsquare():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', k, l)
X = BlockDiagMatrix(A, B)
assert X.shape == (n + k, m + l)
assert X.shape == (n + k, m + l)
assert X.rowblocksizes == [n, k]
assert X.colblocksizes == [m, l]
C = MatrixSymbol('C', n, m)
D = MatrixSymbol('D', k, l)
Y = BlockDiagMatrix(C, D)
assert block_collapse(X + Y) == BlockDiagMatrix(A + C, B + D)
assert block_collapse(X * Y.T) == BlockDiagMatrix(A * C.T, B * D.T)
raises(NonInvertibleMatrixError, lambda: BlockDiagMatrix(A, C.T).inverse())
def test_BlockMatrix():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, k)
C = MatrixSymbol('C', l, m)
D = MatrixSymbol('D', l, k)
M = MatrixSymbol('M', m + k, p)
N = MatrixSymbol('N', l + n, k + m)
X = BlockMatrix(Matrix([[A, B], [C, D]]))
assert X.__class__(*X.args) == X
# block_collapse does nothing on normal inputs
E = MatrixSymbol('E', n, m)
assert block_collapse(A + 2*E) == A + 2*E
F = MatrixSymbol('F', m, m)
assert block_collapse(E.T*A*F) == E.T*A*F
assert X.shape == (l + n, k + m)
assert X.blockshape == (2, 2)
assert transpose(X) == BlockMatrix(Matrix([[A.T, C.T], [B.T, D.T]]))
assert transpose(X).shape == X.shape[::-1]
# Test that BlockMatrices and MatrixSymbols can still mix
assert (X*M).is_MatMul
assert X._blockmul(M).is_MatMul
assert (X*M).shape == (n + l, p)
assert (X + N).is_MatAdd
assert X._blockadd(N).is_MatAdd
assert (X + N).shape == X.shape
E = MatrixSymbol('E', m, 1)
F = MatrixSymbol('F', k, 1)
Y = BlockMatrix(Matrix([[E], [F]]))
assert (X*Y).shape == (l + n, 1)
assert block_collapse(X*Y).blocks[0, 0] == A*E + B*F
assert block_collapse(X*Y).blocks[1, 0] == C*E + D*F
# block_collapse passes down into container objects, transposes, and inverse
assert block_collapse(transpose(X*Y)) == transpose(block_collapse(X*Y))
assert block_collapse(Tuple(X*Y, 2*X)) == (
block_collapse(X*Y), block_collapse(2*X))
# Make sure that MatrixSymbols will enter 1x1 BlockMatrix if it simplifies
Ab = BlockMatrix([[A]])
Z = MatrixSymbol('Z', *A.shape)
assert block_collapse(Ab + Z) == A + Z
def test_block_lu_decomposition():
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, n)
D = MatrixSymbol('D', m, m)
X = BlockMatrix([[A, B], [C, D]])
#LDU decomposition
L, D, U = X.LDUdecomposition()
assert block_collapse(L*D*U) == X
#UDL decomposition
U, D, L = X.UDLdecomposition()
assert block_collapse(U*D*L) == X
#LU decomposition
L, U = X.LUdecomposition()
assert block_collapse(L*U) == X
def test_BlockMatrix_3x3_symbolic():
# Only test one of these, instead of all permutations, because it's slow
rowblocksizes = (n, m, k)
colblocksizes = (m, k, n)
K = BlockMatrix([
[MatrixSymbol('M%s%s' % (rows, cols), rows, cols) for cols in colblocksizes]
for rows in rowblocksizes
])
collapse = block_collapse(K.I)
assert isinstance(collapse, BlockMatrix)
def test_BlockMatrix_2x2_inverse_numeric():
"""Test 2x2 block matrix inversion numerically for all 4 formulas"""
M = Matrix([[1, 2], [3, 4]])
# rank deficient matrices that have full rank when two of them combined
D1 = Matrix([[1, 2], [2, 4]])
D2 = Matrix([[1, 3], [3, 9]])
D3 = Matrix([[1, 4], [4, 16]])
assert D1.rank() == D2.rank() == D3.rank() == 1
assert (D1 + D2).rank() == (D2 + D3).rank() == (D3 + D1).rank() == 2
# Only A is invertible
K = BlockMatrix([[M, D1], [D2, D3]])
assert block_collapse(K.inv()).as_explicit() == K.as_explicit().inv()
# Only B is invertible
K = BlockMatrix([[D1, M], [D2, D3]])
assert block_collapse(K.inv()).as_explicit() == K.as_explicit().inv()
# Only C is invertible
K = BlockMatrix([[D1, D2], [M, D3]])
assert block_collapse(K.inv()).as_explicit() == K.as_explicit().inv()
# Only D is invertible
K = BlockMatrix([[D1, D2], [D3, M]])
assert block_collapse(K.inv()).as_explicit() == K.as_explicit().inv()
def test_issue_21866():
n = 10
I = Identity(n)
O = ZeroMatrix(n, n)
A = BlockMatrix([[ I, O, O, O ],
[ O, I, O, O ],
[ O, O, I, O ],
[ I, O, O, I ]])
Ainv = block_collapse(A.inv())
AinvT = BlockMatrix([[ I, O, O, O ],
[ O, I, O, O ],
[ O, O, I, O ],
[ -I, O, O, I ]])
assert Ainv == AinvT
def test_issue_2460():
bdm1 = BlockDiagMatrix(Matrix([i]), Matrix([j]))
bdm2 = BlockDiagMatrix(Matrix([k]), Matrix([l]))
assert block_collapse(bdm1 + bdm2) == BlockDiagMatrix(Matrix([i + k]), Matrix([j + l]))
def test_issue_17624():
a = MatrixSymbol("a", 2, 2)
z = ZeroMatrix(2, 2)
b = BlockMatrix([[a, z], [z, z]])
assert block_collapse(b * b) == BlockMatrix([[a**2, z], [z, z]])
assert block_collapse(b * b * b) == BlockMatrix([[a**3, z], [z, z]])
def test_block_collapse_explicit_matrices():
A = Matrix([[1, 2], [3, 4]])
assert block_collapse(BlockMatrix([[A]])) == A
A = ImmutableSparseMatrix([[1, 2], [3, 4]])
assert block_collapse(BlockMatrix([[A]])) == A