def test_transpose():
Sq = MatrixSymbol('Sq', n, n)
assert transpose(A) == Transpose(A)
assert Transpose(A).shape == (m, n)
assert Transpose(A * B).shape == (l, n)
assert transpose(Transpose(A)) == A
assert isinstance(Transpose(Transpose(A)), Transpose)
assert adjoint(Transpose(A)) == Adjoint(Transpose(A))
assert conjugate(Transpose(A)) == Adjoint(A)
assert Transpose(eye(3)).doit() == eye(3)
assert Transpose(S(5)).doit() == S(5)
assert Transpose(Matrix([[1, 2], [3, 4]])).doit() == Matrix([[1, 3],
[2, 4]])
assert transpose(trace(Sq)) == trace(Sq)
assert trace(Transpose(Sq)) == trace(Sq)
assert Transpose(Sq)[0, 1] == Sq[1, 0]
assert Transpose(A * B).doit() == Transpose(B) * Transpose(A)
def test_BlockDiagMatrix_trace():
assert trace(BlockDiagMatrix()) == 0
assert trace(BlockDiagMatrix(ZeroMatrix(n, n))) == 0
A = MatrixSymbol('A', n, n)
assert trace(BlockDiagMatrix(A)) == trace(A)
B = MatrixSymbol('B', m, m)
assert trace(BlockDiagMatrix(A, B)) == trace(A) + trace(B)
# non-square blocks
C = MatrixSymbol('C', m, n)
D = MatrixSymbol('D', n, m)
assert isinstance(trace(BlockDiagMatrix(C, D)), Trace)
def test_issue_9817_simplify():
# simplify on trace of substituted explicit quadratic form of matrix
# expressions (a scalar) should return without errors (AttributeError)
# See issue #9817 and #9190 for the original bug more discussion on this
from sympy.matrices.expressions import Identity, trace
v = MatrixSymbol('v', 3, 1)
A = MatrixSymbol('A', 3, 3)
x = Matrix([i + 1 for i in range(3)])
X = Identity(3)
quadratic = v.T * A * v
assert simplify((trace(quadratic.as_explicit())).xreplace({v:x, A:X})) == 14
def test_adjoint():
Sq = MatrixSymbol("Sq", n, n)
assert Adjoint(A).shape == (m, n)
assert Adjoint(A * B).shape == (l, n)
assert adjoint(Adjoint(A)) == A
assert isinstance(Adjoint(Adjoint(A)), Adjoint)
assert conjugate(Adjoint(A)) == Transpose(A)
assert transpose(Adjoint(A)) == Adjoint(Transpose(A))
assert Adjoint(eye(3)).doit() == eye(3)
assert Adjoint(S(5)).doit() == S(5)
assert Adjoint(Matrix([[1, 2], [3, 4]])).doit() == Matrix([[1, 3], [2, 4]])
assert adjoint(trace(Sq)) == conjugate(trace(Sq))
assert trace(adjoint(Sq)) == conjugate(trace(Sq))
assert Adjoint(Sq)[0, 1] == conjugate(Sq[1, 0])
assert Adjoint(A * B).doit() == Adjoint(B) * Adjoint(A)
def test_adjoint():
Sq = MatrixSymbol('Sq', n, n)
assert Adjoint(A).shape == (m, n)
assert Adjoint(A*B).shape == (l, n)
assert adjoint(Adjoint(A)) == A
assert isinstance(Adjoint(Adjoint(A)), Adjoint)
assert conjugate(Adjoint(A)) == Transpose(A)
assert transpose(Adjoint(A)) == Adjoint(Transpose(A))
assert Adjoint(eye(3)).doit() == eye(3)
assert Adjoint(S(5)).doit() == S(5)
assert Adjoint(Matrix([[1, 2], [3, 4]])).doit() == Matrix([[1, 3], [2, 4]])
assert adjoint(trace(Sq)) == conjugate(trace(Sq))
assert trace(adjoint(Sq)) == conjugate(trace(Sq))
assert Adjoint(Sq)[0, 1] == conjugate(Sq[1, 0])
assert Adjoint(A*B).doit() == Adjoint(B) * Adjoint(A)
def test_KroneckerProduct_trace():
kp = kronecker_product(W, Z)
assert trace(kp) == trace(W)*trace(Z)
def test_rewrite():
assert isinstance(trace(A).rewrite(Sum), Sum)
def test_Trace():
assert isinstance(Trace(A), Trace)
assert not isinstance(Trace(A), MatrixExpr)
raises(ShapeError, lambda: Trace(C))
assert trace(eye(3)) == 3
assert trace(Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])) == 15
assert adjoint(Trace(A)) == trace(Adjoint(A))
assert conjugate(Trace(A)) == trace(Adjoint(A))
assert transpose(Trace(A)) == Trace(A)
A / Trace(A) # Make sure this is possible
# Some easy simplifications
assert trace(Identity(5)) == 5
assert trace(ZeroMatrix(5, 5)) == 0
assert trace(2 * A * B) == 2 * trace(A * B)
assert trace(A.T) == trace(A)
i, j = symbols('i j')
F = FunctionMatrix(3, 3, Lambda((i, j), i + j))
assert trace(F) == (0 + 0) + (1 + 1) + (2 + 2)
raises(TypeError, lambda: Trace(S.One))
assert Trace(A).arg is A
assert str(trace(A)) == str(Trace(A).doit())
def test_trace_constant_factor():
# Issue 9052: LHS gives Trace(MatMul(A))
assert trace(2 * A) == 2 * Trace(A)
X = ImmutableMatrix([[1, 2], [3, 4]])
assert trace(MatMul(2, X)) == 10
def test_KroneckerProduct_trace():
kp = kronecker_product(W, Z)
assert trace(kp) == trace(W) * trace(Z)
def test_trace_constant_factor():
# Issue 9052: gave 2*Trace(MatMul(A)) instead of 2*Trace(A)
assert trace(2*A) == 2*Trace(A)
X = ImmutableMatrix([[1, 2], [3, 4]])
assert trace(MatMul(2, X)) == 10
def test_Trace():
assert isinstance(Trace(A), Trace)
assert not isinstance(Trace(A), MatrixExpr)
raises(ShapeError, lambda: Trace(C))
assert trace(eye(3)) == 3
assert trace(Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])) == 15
assert adjoint(Trace(A)) == trace(Adjoint(A))
assert conjugate(Trace(A)) == trace(Adjoint(A))
assert transpose(Trace(A)) == Trace(A)
A / Trace(A) # Make sure this is possible
# Some easy simplifications
assert trace(Identity(5)) == 5
assert trace(ZeroMatrix(5, 5)) == 0
assert trace(OneMatrix(1, 1)) == 1
assert trace(OneMatrix(2, 2)) == 2
assert trace(OneMatrix(n, n)) == n
assert trace(2*A*B) == 2*Trace(A*B)
assert trace(A.T) == trace(A)
i, j = symbols('i j')
F = FunctionMatrix(3, 3, Lambda((i, j), i + j))
assert trace(F) == (0 + 0) + (1 + 1) + (2 + 2)
raises(TypeError, lambda: Trace(S.One))
assert Trace(A).arg is A
assert str(trace(A)) == str(Trace(A).doit())
assert Trace(A).is_commutative is True
def test_trace_rewrite():
assert trace(A).rewrite(Sum) == Sum(A[i, i], (i, 0, n - 1))
assert trace(eye(3)).rewrite(Sum) == 3
def test_BlockMatrix_trace():
A, B, C, D = [MatrixSymbol(s, 3, 3) for s in 'ABCD']
X = BlockMatrix([[A, B], [C, D]])
assert trace(X) == trace(A) + trace(D)
assert trace(BlockMatrix([ZeroMatrix(n, n)])) == 0
def test_BlockMatrix_trace():
A, B, C, D = [MatrixSymbol(s, 3, 3) for s in "ABCD"]
X = BlockMatrix([[A, B], [C, D]])
assert trace(X) == trace(A) + trace(D)
def test_BlockMatrix_trace():
A, B, C, D = map(lambda s: MatrixSymbol(s, 3, 3), "ABCD")
X = BlockMatrix([[A, B], [C, D]])
assert trace(X) == trace(A) + trace(D)
def test_trace_constant_factor():
# Issue 9052: LHS gives Trace(MatMul(A))
assert trace(2*A) == 2*Trace(A)
X = ImmutableMatrix([[1, 2], [3, 4]])
assert trace(MatMul(2, X)) == 10
def test_Trace_A_plus_B():
assert trace(A + B) == Trace(A) + Trace(B)
assert Trace(A + B).arg == MatAdd(A, B)
assert Trace(A + B).doit() == Trace(A) + Trace(B)
def test_BlockMatrix_trace():
A, B, C, D = map(lambda s: MatrixSymbol(s, 3, 3), 'ABCD')
X = BlockMatrix([[A, B], [C, D]])
assert trace(X) == trace(A) + trace(D)
def test_rewrite():
assert isinstance(trace(A).rewrite(Sum), Sum)
def test_Trace_A_plus_B():
assert trace(A + B) == Trace(A) + Trace(B)
assert Trace(A + B).arg == MatAdd(A, B)
assert Trace(A + B).doit() == Trace(A) + Trace(B)
def test_trace_constant_factor():
# Issue 9052: gave 2*Trace(MatMul(A)) instead of 2*Trace(A)
assert trace(2 * A) == 2 * Trace(A)
X = ImmutableMatrix([[1, 2], [3, 4]])
assert trace(MatMul(2, X)) == 10
def test_BlockMatrix_trace():
A, B, C, D = [MatrixSymbol(s, 3, 3) for s in 'ABCD']
X = BlockMatrix([[A, B], [C, D]])
assert trace(X) == trace(A) + trace(D)
