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_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 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_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 (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_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],
])