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_eval_determinant():
assert det(Identity(n)) == 1
assert det(ZeroMatrix(n, n)) == 0
assert det(OneMatrix(n, n)) == Determinant(OneMatrix(n, n))
assert det(OneMatrix(1, 1)) == 1
assert det(OneMatrix(2, 2)) == 0
assert det(Transpose(A)) == det(A)
def test_MatrixPermute_doit():
p = Permutation(0, 1, 2)
A = MatrixSymbol('A', 3, 3)
assert MatrixPermute(A, p).doit() == MatrixPermute(A, p)
p = Permutation(0, size=3)
A = MatrixSymbol('A', 3, 3)
assert MatrixPermute(A, p).doit().as_explicit() == \
MatrixPermute(A, p).as_explicit()
p = Permutation(0, 1, 2)
A = Identity(3)
assert MatrixPermute(A, p, 0).doit().as_explicit() == \
MatrixPermute(A, p, 0).as_explicit()
assert MatrixPermute(A, p, 1).doit().as_explicit() == \
MatrixPermute(A, p, 1).as_explicit()
A = ZeroMatrix(3, 3)
assert MatrixPermute(A, p).doit() == A
A = OneMatrix(3, 3)
assert MatrixPermute(A, p).doit() == A
A = MatrixSymbol('A', 4, 4)
p1 = Permutation(0, 1, 2, 3)
p2 = Permutation(0, 2, 3, 1)
expr = MatrixPermute(MatrixPermute(A, p1, 0), p2, 0)
assert expr.as_explicit() == expr.doit().as_explicit()
expr = MatrixPermute(MatrixPermute(A, p1, 1), p2, 1)
assert expr.as_explicit() == expr.doit().as_explicit()
def _(expr: ArrayContraction):
expr = expr.flatten_contraction_of_diagonal()
expr = identify_removable_identity_matrices(expr)
expr = expr.split_multiple_contractions()
expr = identify_hadamard_products(expr)
if not isinstance(expr, ArrayContraction):
return _array2matrix(expr)
subexpr = expr.expr
contraction_indices: tTuple[tTuple[int]] = expr.contraction_indices
if contraction_indices == ((0, ), (1, )) or (contraction_indices == (
(0, ), ) and subexpr.shape[1] == 1) or (contraction_indices == (
(1, ), ) and subexpr.shape[0] == 1):
shape = subexpr.shape
subexpr = _array2matrix(subexpr)
if isinstance(subexpr, MatrixExpr):
return OneMatrix(1, shape[0]) * subexpr * OneMatrix(shape[1], 1)
if isinstance(subexpr, ArrayTensorProduct):
newexpr = _array_contraction(_array2matrix(subexpr),
*contraction_indices)
contraction_indices = newexpr.contraction_indices
if any(i > 2 for i in newexpr.subranks):
addends = _array_add(*[
_a2m_tensor_product(*j) for j in itertools.product(*[
i.args if isinstance(i, ArrayAdd) else [i]
for i in expr.expr.args
])
])
newexpr = _array_contraction(addends, *contraction_indices)
if isinstance(newexpr, ArrayAdd):
ret = _array2matrix(newexpr)
return ret
assert isinstance(newexpr, ArrayContraction)
ret = _support_function_tp1_recognize(contraction_indices,
list(newexpr.expr.args))
return ret
elif not isinstance(subexpr, _CodegenArrayAbstract):
ret = _array2matrix(subexpr)
if isinstance(ret, MatrixExpr):
assert expr.contraction_indices == ((0, 1), )
return _a2m_trace(ret)
else:
return _array_contraction(ret, *expr.contraction_indices)
def test_arrayexpr_convert_array_to_matrix2():
cg = _array_contraction(_array_tensor_product(M, N), (1, 3))
assert convert_array_to_matrix(cg) == M * N.T
cg = PermuteDims(_array_tensor_product(M, N), Permutation([0, 1, 3, 2]))
assert convert_array_to_matrix(cg) == _array_tensor_product(M, N.T)
cg = _array_tensor_product(M, PermuteDims(N, Permutation([1, 0])))
assert convert_array_to_matrix(cg) == _array_tensor_product(M, N.T)
cg = _array_contraction(
PermuteDims(
_array_tensor_product(M, N, P, Q), Permutation([0, 2, 3, 1, 4, 5, 7, 6])),
(1, 2), (3, 5)
)
assert convert_array_to_matrix(cg) == _array_tensor_product(M * P.T * Trace(N), Q.T)
cg = _array_contraction(
_array_tensor_product(M, N, P, PermuteDims(Q, Permutation([1, 0]))),
(1, 5), (2, 3)
)
assert convert_array_to_matrix(cg) == _array_tensor_product(M * P.T * Trace(N), Q.T)
cg = _array_tensor_product(M, PermuteDims(N, [1, 0]))
assert convert_array_to_matrix(cg) == _array_tensor_product(M, N.T)
cg = _array_tensor_product(PermuteDims(M, [1, 0]), PermuteDims(N, [1, 0]))
assert convert_array_to_matrix(cg) == _array_tensor_product(M.T, N.T)
cg = _array_tensor_product(PermuteDims(N, [1, 0]), PermuteDims(M, [1, 0]))
assert convert_array_to_matrix(cg) == _array_tensor_product(N.T, M.T)
cg = _array_contraction(M, (0,), (1,))
assert convert_array_to_matrix(cg) == OneMatrix(1, k)*M*OneMatrix(k, 1)
cg = _array_contraction(x, (0,), (1,))
assert convert_array_to_matrix(cg) == OneMatrix(1, k)*x
Xm = MatrixSymbol("Xm", m, n)
cg = _array_contraction(Xm, (0,), (1,))
assert convert_array_to_matrix(cg) == OneMatrix(1, m)*Xm*OneMatrix(n, 1)
def test_canonicalize():
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
expr = HadamardProduct(X, check=False)
assert isinstance(expr, HadamardProduct)
expr2 = expr.doit() # unpack is called
assert isinstance(expr2, MatrixSymbol)
Z = ZeroMatrix(2, 2)
U = OneMatrix(2, 2)
assert HadamardProduct(Z, X).doit() == Z
assert HadamardProduct(U, X, X, U).doit() == HadamardPower(X, 2)
assert HadamardProduct(X, U, Y).doit() == HadamardProduct(X, Y)
assert HadamardProduct(X, Z, U, Y).doit() == Z
def test_NumPyPrinter():
from sympy.core.function import Lambda
from sympy.matrices.expressions.adjoint import Adjoint
from sympy.matrices.expressions.diagonal import (DiagMatrix,
DiagonalMatrix,
DiagonalOf)
from sympy.matrices.expressions.funcmatrix import FunctionMatrix
from sympy.matrices.expressions.hadamard import HadamardProduct
from sympy.matrices.expressions.kronecker import KroneckerProduct
from sympy.matrices.expressions.special import (OneMatrix, ZeroMatrix)
from sympy.abc import a, b
p = NumPyPrinter()
assert p.doprint(sign(x)) == 'numpy.sign(x)'
A = MatrixSymbol("A", 2, 2)
B = MatrixSymbol("B", 2, 2)
C = MatrixSymbol("C", 1, 5)
D = MatrixSymbol("D", 3, 4)
assert p.doprint(A**(-1)) == "numpy.linalg.inv(A)"
assert p.doprint(A**5) == "numpy.linalg.matrix_power(A, 5)"
assert p.doprint(Identity(3)) == "numpy.eye(3)"
u = MatrixSymbol('x', 2, 1)
v = MatrixSymbol('y', 2, 1)
assert p.doprint(MatrixSolve(A, u)) == 'numpy.linalg.solve(A, x)'
assert p.doprint(MatrixSolve(A, u) + v) == 'numpy.linalg.solve(A, x) + y'
assert p.doprint(ZeroMatrix(2, 3)) == "numpy.zeros((2, 3))"
assert p.doprint(OneMatrix(2, 3)) == "numpy.ones((2, 3))"
assert p.doprint(FunctionMatrix(4, 5, Lambda((a, b), a + b))) == \
"numpy.fromfunction(lambda a, b: a + b, (4, 5))"
assert p.doprint(HadamardProduct(A, B)) == "numpy.multiply(A, B)"
assert p.doprint(KroneckerProduct(A, B)) == "numpy.kron(A, B)"
assert p.doprint(Adjoint(A)) == "numpy.conjugate(numpy.transpose(A))"
assert p.doprint(DiagonalOf(A)) == "numpy.reshape(numpy.diag(A), (-1, 1))"
assert p.doprint(DiagMatrix(C)) == "numpy.diagflat(C)"
assert p.doprint(DiagonalMatrix(D)) == "numpy.multiply(D, numpy.eye(3, 4))"
# Workaround for numpy negative integer power errors
assert p.doprint(x**-1) == 'x**(-1.0)'
assert p.doprint(x**-2) == 'x**(-2.0)'
expr = Pow(2, -1, evaluate=False)
assert p.doprint(expr) == "2**(-1.0)"
assert p.doprint(S.Exp1) == 'numpy.e'
assert p.doprint(S.Pi) == 'numpy.pi'
assert p.doprint(S.EulerGamma) == 'numpy.euler_gamma'
assert p.doprint(S.NaN) == 'numpy.nan'
assert p.doprint(S.Infinity) == 'numpy.PINF'
assert p.doprint(S.NegativeInfinity) == 'numpy.NINF'
def test_matrix_derivative_by_scalar():
assert A.diff(i) == ZeroMatrix(k, k)
assert (A*(X + B)*c).diff(i) == ZeroMatrix(k, 1)
assert x.diff(i) == ZeroMatrix(k, 1)
assert (x.T*y).diff(i) == ZeroMatrix(1, 1)
assert (x*x.T).diff(i) == ZeroMatrix(k, k)
assert (x + y).diff(i) == ZeroMatrix(k, 1)
assert hadamard_power(x, 2).diff(i) == ZeroMatrix(k, 1)
assert hadamard_power(x, i).diff(i).dummy_eq(
HadamardProduct(x.applyfunc(log), HadamardPower(x, i)))
assert hadamard_product(x, y).diff(i) == ZeroMatrix(k, 1)
assert hadamard_product(i*OneMatrix(k, 1), x, y).diff(i) == hadamard_product(x, y)
assert (i*x).diff(i) == x
assert (sin(i)*A*B*x).diff(i) == cos(i)*A*B*x
assert x.applyfunc(sin).diff(i) == ZeroMatrix(k, 1)
assert Trace(i**2*X).diff(i) == 2*i*Trace(X)
mu = symbols("mu")
expr = (2*mu*x)
assert expr.diff(x) == 2*mu*Identity(k)
def test_hadamard():
m, n, p = symbols('m, n, p', integer=True)
A = MatrixSymbol('A', m, n)
B = MatrixSymbol('B', m, n)
C = MatrixSymbol('C', m, p)
X = MatrixSymbol('X', m, m)
I = Identity(m)
with raises(TypeError):
hadamard_product()
assert hadamard_product(A) == A
assert isinstance(hadamard_product(A, B), HadamardProduct)
assert hadamard_product(A, B).doit() == hadamard_product(A, B)
with raises(ShapeError):
hadamard_product(A, C)
hadamard_product(A, I)
assert hadamard_product(X, I) == X
assert isinstance(hadamard_product(X, I), MatrixSymbol)
a = MatrixSymbol("a", k, 1)
expr = MatAdd(ZeroMatrix(k, 1), OneMatrix(k, 1))
expr = HadamardProduct(expr, a)
assert expr.doit() == a
def test_arrayexpr_convert_array_to_matrix_remove_trivial_dims():
# Tensor Product:
assert _remove_trivial_dims(_array_tensor_product(a, b)) == (a * b.T, [1, 3])
assert _remove_trivial_dims(_array_tensor_product(a.T, b)) == (a * b.T, [0, 3])
assert _remove_trivial_dims(_array_tensor_product(a, b.T)) == (a * b.T, [1, 2])
assert _remove_trivial_dims(_array_tensor_product(a.T, b.T)) == (a * b.T, [0, 2])
assert _remove_trivial_dims(_array_tensor_product(I, a.T, b.T)) == (_array_tensor_product(I, a * b.T), [2, 4])
assert _remove_trivial_dims(_array_tensor_product(a.T, I, b.T)) == (_array_tensor_product(a.T, I, b.T), [])
assert _remove_trivial_dims(_array_tensor_product(a, I)) == (_array_tensor_product(a, I), [])
assert _remove_trivial_dims(_array_tensor_product(I, a)) == (_array_tensor_product(I, a), [])
assert _remove_trivial_dims(_array_tensor_product(a.T, b.T, c, d)) == (
_array_tensor_product(a * b.T, c * d.T), [0, 2, 5, 7])
assert _remove_trivial_dims(_array_tensor_product(a.T, I, b.T, c, d, I)) == (
_array_tensor_product(a.T, I, b*c.T, d, I), [4, 7])
# Addition:
cg = ArrayAdd(_array_tensor_product(a, b), _array_tensor_product(c, d))
assert _remove_trivial_dims(cg) == (a * b.T + c * d.T, [1, 3])
# Permute Dims:
cg = PermuteDims(_array_tensor_product(a, b), Permutation(3)(1, 2))
assert _remove_trivial_dims(cg) == (a * b.T, [2, 3])
cg = PermuteDims(_array_tensor_product(a, I, b), Permutation(5)(1, 2, 3, 4))
assert _remove_trivial_dims(cg) == (cg, [])
cg = PermuteDims(_array_tensor_product(I, b, a), Permutation(5)(1, 2, 4, 5, 3))
assert _remove_trivial_dims(cg) == (PermuteDims(_array_tensor_product(I, b * a.T), [0, 2, 3, 1]), [4, 5])
# Diagonal:
cg = _array_diagonal(_array_tensor_product(M, a), (1, 2))
assert _remove_trivial_dims(cg) == (cg, [])
# Contraction:
cg = _array_contraction(_array_tensor_product(M, a), (1, 2))
assert _remove_trivial_dims(cg) == (cg, [])
# A few more cases to test the removal and shift of nested removed axes
# with array contractions and array diagonals:
tp = _array_tensor_product(
OneMatrix(1, 1),
M,
x,
OneMatrix(1, 1),
Identity(1),
)
expr = _array_contraction(tp, (1, 8))
rexpr, removed = _remove_trivial_dims(expr)
assert removed == [0, 5, 6, 7]
expr = _array_contraction(tp, (1, 8), (3, 4))
rexpr, removed = _remove_trivial_dims(expr)
assert removed == [0, 3, 4, 5]
expr = _array_diagonal(tp, (1, 8))
rexpr, removed = _remove_trivial_dims(expr)
assert removed == [0, 5, 6, 7, 8]
expr = _array_diagonal(tp, (1, 8), (3, 4))
rexpr, removed = _remove_trivial_dims(expr)
assert removed == [0, 3, 4, 5, 6]
expr = _array_diagonal(_array_contraction(_array_tensor_product(A, x, I, I1), (1, 2, 5)), (1, 4))
rexpr, removed = _remove_trivial_dims(expr)
assert removed == [2, 3]
cg = _array_diagonal(_array_tensor_product(PermuteDims(_array_tensor_product(x, I1), Permutation(1, 2, 3)), (x.T*x).applyfunc(sqrt)), (2, 4), (3, 5))
rexpr, removed = _remove_trivial_dims(cg)
assert removed == [1, 2]
# Contractions with identity matrices need to be followed by a permutation
# in order
cg = _array_contraction(_array_tensor_product(A, B, C, M, I), (1, 8))
ret, removed = _remove_trivial_dims(cg)
assert ret == PermuteDims(_array_tensor_product(A, B, C, M), [0, 2, 3, 4, 5, 6, 7, 1])
assert removed == []
cg = _array_contraction(_array_tensor_product(A, B, C, M, I), (1, 8), (3, 4))
ret, removed = _remove_trivial_dims(cg)
assert ret == PermuteDims(_array_contraction(_array_tensor_product(A, B, C, M), (3, 4)), [0, 2, 3, 4, 5, 1])
assert removed == []
# Trivial matrices are sometimes inserted into MatMul expressions:
cg = _array_tensor_product(b*b.T, a.T*a)
ret, removed = _remove_trivial_dims(cg)
assert ret == b*a.T*a*b.T
assert removed == [2, 3]
Xs = ArraySymbol("X", (3, 2, k))
cg = _array_tensor_product(M, Xs, b.T*c, a*a.T, b*b.T, c.T*d)
ret, removed = _remove_trivial_dims(cg)
assert ret == _array_tensor_product(M, Xs, a*b.T*c*c.T*d*a.T, b*b.T)
assert removed == [5, 6, 11, 12]
cg = _array_diagonal(_array_tensor_product(I, I1, x), (1, 4), (3, 5))
assert _remove_trivial_dims(cg) == (PermuteDims(_array_diagonal(_array_tensor_product(I, x), (1, 2)), Permutation(1, 2)), [1])
expr = _array_diagonal(_array_tensor_product(x, I, y), (0, 2))
assert _remove_trivial_dims(expr) == (PermuteDims(_array_tensor_product(DiagMatrix(x), y), [1, 2, 3, 0]), [0])
expr = _array_diagonal(_array_tensor_product(x, I, y), (0, 2), (3, 4))
assert _remove_trivial_dims(expr) == (expr, [])
def test_one_matrix_creation():
assert OneMatrix(2, 2)
assert OneMatrix(0, 0)
assert Eq(OneMatrix(1, 1), Identity(1))
raises(ValueError, lambda: OneMatrix(-1, 2))
raises(ValueError, lambda: OneMatrix(2.0, 2))
raises(ValueError, lambda: OneMatrix(2j, 2))
raises(ValueError, lambda: OneMatrix(2, -1))
raises(ValueError, lambda: OneMatrix(2, 2.0))
raises(ValueError, lambda: OneMatrix(2, 2j))
n = symbols('n')
assert OneMatrix(n, n)
n = symbols('n', integer=False)
raises(ValueError, lambda: OneMatrix(n, n))
n = symbols('n', negative=True)
raises(ValueError, lambda: OneMatrix(n, n))
def test_OneMatrix_mul():
n, m, k = symbols('n m k', integer=True)
w = MatrixSymbol('w', n, 1)
assert OneMatrix(n, m) * OneMatrix(m, k) == OneMatrix(n, k) * m
assert w * OneMatrix(1, 1) == w
assert OneMatrix(1, 1) * w.T == w.T
def test_OneMatrix_doit():
n = symbols('n', integer=True)
Unn = OneMatrix(Add(n, n, evaluate=False), n)
assert isinstance(Unn.rows, Add)
assert Unn.doit() == OneMatrix(2 * n, n)
assert isinstance(Unn.doit().rows, Mul)
def test_OneMatrix():
n, m = symbols('n m', integer=True)
A = MatrixSymbol('A', n, m)
a = MatrixSymbol('a', n, 1)
U = OneMatrix(n, m)
assert U.shape == (n, m)
assert isinstance(A + U, Add)
assert U.transpose() == OneMatrix(m, n)
assert U.conjugate() == U
assert OneMatrix(n, n)**0 == Identity(n)
with raises(NonSquareMatrixError):
U**0
with raises(NonSquareMatrixError):
U**1
with raises(NonSquareMatrixError):
U**2
with raises(ShapeError):
a + U
U = OneMatrix(n, n)
assert U[1, 2] == 1
U = OneMatrix(2, 3)
assert U.as_explicit() == ImmutableDenseMatrix.ones(2, 3)
def test_matrix_expression_from_index_summation():
from sympy.abc import a,b,c,d
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
C = MatrixSymbol("C", k, k)
w1 = MatrixSymbol("w1", k, 1)
i0, i1, i2, i3, i4 = symbols("i0:5", cls=Dummy)
expr = Sum(W[a,b]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 0, m-1))
assert MatrixExpr.from_index_summation(expr, a) == W*X*Z
expr = Sum(W.T[b,a]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 0, m-1))
assert MatrixExpr.from_index_summation(expr, a) == W*X*Z
expr = Sum(A[b, a]*B[b, c]*C[c, d], (b, 0, k-1), (c, 0, k-1))
assert MatrixSymbol.from_index_summation(expr, a) == A.T*B*C
expr = Sum(A[b, a]*B[c, b]*C[c, d], (b, 0, k-1), (c, 0, k-1))
assert MatrixSymbol.from_index_summation(expr, a) == A.T*B.T*C
expr = Sum(C[c, d]*A[b, a]*B[c, b], (b, 0, k-1), (c, 0, k-1))
assert MatrixSymbol.from_index_summation(expr, a) == A.T*B.T*C
expr = Sum(A[a, b] + B[a, b], (a, 0, k-1), (b, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == OneMatrix(1, k)*A*OneMatrix(k, 1) + OneMatrix(1, k)*B*OneMatrix(k, 1)
expr = Sum(A[a, b]**2, (a, 0, k - 1), (b, 0, k - 1))
assert MatrixExpr.from_index_summation(expr, a) == Trace(A * A.T)
expr = Sum(A[a, b]**3, (a, 0, k - 1), (b, 0, k - 1))
assert MatrixExpr.from_index_summation(expr, a) == Trace(HadamardPower(A.T, 2) * A)
expr = Sum((A[a, b] + B[a, b])*C[b, c], (b, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == (A+B)*C
expr = Sum((A[a, b] + B[b, a])*C[b, c], (b, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == (A+B.T)*C
expr = Sum(A[a, b]*A[b, c]*A[c, d], (b, 0, k-1), (c, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == A**3
expr = Sum(A[a, b]*A[b, c]*B[c, d], (b, 0, k-1), (c, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == A**2*B
# Parse the trace of a matrix:
expr = Sum(A[a, a], (a, 0, k-1))
assert MatrixExpr.from_index_summation(expr, None) == trace(A)
expr = Sum(A[a, a]*B[b, c]*C[c, d], (a, 0, k-1), (c, 0, k-1))
assert MatrixExpr.from_index_summation(expr, b) == trace(A)*B*C
# Check wrong sum ranges (should raise an exception):
## Case 1: 0 to m instead of 0 to m-1
expr = Sum(W[a,b]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 0, m))
raises(ValueError, lambda: MatrixExpr.from_index_summation(expr, a))
## Case 2: 1 to m-1 instead of 0 to m-1
expr = Sum(W[a,b]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 1, m-1))
raises(ValueError, lambda: MatrixExpr.from_index_summation(expr, a))
# Parse nested sums:
expr = Sum(A[a, b]*Sum(B[b, c]*C[c, d], (c, 0, k-1)), (b, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == A*B*C
# Test Kronecker delta:
expr = Sum(A[a, b]*KroneckerDelta(b, c)*B[c, d], (b, 0, k-1), (c, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == A*B
expr = Sum(KroneckerDelta(i1, m)*KroneckerDelta(i2, n)*A[i, i1]*A[j, i2], (i1, 0, k-1), (i2, 0, k-1))
assert MatrixExpr.from_index_summation(expr, m) == ArrayTensorProduct(A.T, A)
# Test numbered indices:
expr = Sum(A[i1, i2]*w1[i2, 0], (i2, 0, k-1))
assert MatrixExpr.from_index_summation(expr, i1) == MatrixElement(A*w1, i1, 0)
expr = Sum(A[i1, i2]*B[i2, 0], (i2, 0, k-1))
assert MatrixExpr.from_index_summation(expr, i1) == MatrixElement(A*B, i1, 0)