def test_entry():
from sympy.concrete import Sum
assert MatPow(A, 1)[0, 0] == A[0, 0]
assert MatPow(C, 0)[0, 0] == 1
assert MatPow(C, 0)[0, 1] == 0
assert isinstance(MatPow(C, 2)[0, 0], Sum)
def test_as_explicit_nonsquare():
A = ImmutableMatrix([[1, 2, 3], [4, 5, 6]])
assert MatPow(A, 1).as_explicit() == A
raises(ShapeError, lambda: MatPow(A, 0).as_explicit())
raises(ShapeError, lambda: MatPow(A, 2).as_explicit())
raises(ShapeError, lambda: MatPow(A, -1).as_explicit())
raises(ValueError, lambda: MatPow(A, pi).as_explicit())
def test_entry_matrix():
X = ImmutableMatrix([[1, 2], [3, 4]])
assert MatPow(X, 0)[0, 0] == 1
assert MatPow(X, 0)[0, 1] == 0
assert MatPow(X, 1)[0, 0] == 1
assert MatPow(X, 1)[0, 1] == 2
assert MatPow(X, 2)[0, 0] == 7
def test_doit_nonsquare():
X = ImmutableMatrix([[1, 2, 3], [4, 5, 6]])
assert MatPow(X, 1).doit() == X
raises(ShapeError, lambda: MatPow(X, 0).doit())
raises(ShapeError, lambda: MatPow(X, 2).doit())
raises(ShapeError, lambda: MatPow(X, -1).doit())
raises(ShapeError, lambda: MatPow(X, pi).doit())
def test_entry_symbol():
from sympy.concrete import Sum
assert MatPow(C, 0)[0, 0] == 1
assert MatPow(C, 0)[0, 1] == 0
assert MatPow(C, 1)[0, 0] == C[0, 0]
assert isinstance(MatPow(C, 2)[0, 0], Sum)
assert isinstance(MatPow(C, n)[0, 0], MatrixElement)
def test_unchanged():
assert unchanged(MatPow, C, 0)
assert unchanged(MatPow, C, 1)
assert unchanged(MatPow, Inverse(C), -1)
assert unchanged(Inverse, MatPow(C, -1), -1)
assert unchanged(MatPow, MatPow(C, -1), -1)
assert unchanged(MatPow, MatPow(C, 1), 1)
def test_combine_powers():
assert (C**1)**1 == C
assert (C**2)**3 == MatPow(C, 6)
assert (C**-2)**-3 == MatPow(C, 6)
assert (C**-1)**-1 == C
assert (((C**2)**3)**4)**5 == MatPow(C, 120)
assert (C**n)**n == C**(n**2)
def test_as_explicit_symbol():
X = MatrixSymbol('X', 2, 2)
assert MatPow(X, 0).as_explicit() == ImmutableMatrix(Identity(2))
assert MatPow(X, 1).as_explicit() == X.as_explicit()
assert MatPow(X, 2).as_explicit() == (X.as_explicit())**2
assert MatPow(X, n).as_explicit() == ImmutableMatrix([
[(X**n)[0, 0], (X**n)[0, 1]],
[(X**n)[1, 0], (X**n)[1, 1]],
])
a = MatrixSymbol("a", 3, 1)
b = MatrixSymbol("b", 3, 1)
c = MatrixSymbol("c", 3, 1)
expr = (a.T * b)**S.Half
assert expr.as_explicit() == Matrix(
[[sqrt(a[0, 0] * b[0, 0] + a[1, 0] * b[1, 0] + a[2, 0] * b[2, 0])]])
expr = c * (a.T * b)**S.Half
m = sqrt(a[0, 0] * b[0, 0] + a[1, 0] * b[1, 0] + a[2, 0] * b[2, 0])
assert expr.as_explicit() == Matrix([[c[0, 0] * m], [c[1, 0] * m],
[c[2, 0] * m]])
expr = (a * b.T)**S.Half
denom = sqrt(a[0, 0] * b[0, 0] + a[1, 0] * b[1, 0] + a[2, 0] * b[2, 0])
expected = (a * b.T).as_explicit() / denom
assert expr.as_explicit() == expected
expr = X**-1
det = X[0, 0] * X[1, 1] - X[1, 0] * X[0, 1]
expected = Matrix([[X[1, 1], -X[0, 1]], [-X[1, 0], X[0, 0]]]) / det
assert expr.as_explicit() == expected
expr = X**m
assert expr.as_explicit() == X.as_explicit()**m
def test_as_explicit_symbol():
X = MatrixSymbol('X', 2, 2)
assert MatPow(X, 0).as_explicit() == ImmutableMatrix(Identity(2))
assert MatPow(X, 1).as_explicit() == X.as_explicit()
assert MatPow(X, 2).as_explicit() == (X.as_explicit())**2
assert MatPow(X, n).as_explicit() == ImmutableMatrix([
[(X**n)[0, 0], (X**n)[0, 1]],
[(X**n)[1, 0], (X**n)[1, 1]],
])
def test_Inverse():
assert Inverse(MatPow(C, 0)).doit() == Identity(n)
assert Inverse(MatPow(C, 1)).doit() == Inverse(C)
assert Inverse(MatPow(C, 2)).doit() == MatPow(C, -2)
assert Inverse(MatPow(C, -1)).doit() == C
assert MatPow(Inverse(C), 0).doit() == Identity(n)
assert MatPow(Inverse(C), 1).doit() == Inverse(C)
assert MatPow(Inverse(C), 2).doit() == MatPow(C, -2)
assert MatPow(Inverse(C), -1).doit() == C
def test_zero_power():
z1 = ZeroMatrix(n, n)
assert MatPow(z1, 3).doit() == z1
raises(ValueError, lambda:MatPow(z1, -1).doit())
assert MatPow(z1, 0).doit() == Identity(n)
assert MatPow(z1, n).doit() == z1
raises(ValueError, lambda:MatPow(z1, -2).doit())
z2 = ZeroMatrix(4, 4)
assert MatPow(z2, n).doit() == z2
raises(ValueError, lambda:MatPow(z2, -3).doit())
assert MatPow(z2, 2).doit() == z2
assert MatPow(z2, 0).doit() == Identity(4)
raises(ValueError, lambda:MatPow(z2, -1).doit())
def test_Trace_doit():
X = Matrix([[1, 2], [3, 4]])
assert Trace(X).doit() == 5
q = MatPow(X, 2)
assert Trace(q).arg == q
assert Trace(q).doit() == 29
assert Trace(q).doit(deep=False).arg == q
def test_doit_args():
A = ImmutableMatrix([[1, 2], [3, 4]])
B = ImmutableMatrix([[2, 3], [4, 5]])
assert MatAdd(A, MatPow(B, 2)).doit() == A + B**2
assert MatAdd(A, MatMul(A, B)).doit() == A + A*B
assert (MatAdd(A, X, MatMul(A, B), Y, MatAdd(2*A, B)).doit() ==
MatAdd(3*A + A*B + B, X, Y))
def test_Trace_doit_deep_False():
X = Matrix([[1, 2], [3, 4]])
q = MatPow(X, 2)
assert Trace(q).doit(deep=False).arg == q
q = MatAdd(X, 2 * X)
assert Trace(q).doit(deep=False).arg == q
q = MatMul(X, 2 * X)
assert Trace(q).doit(deep=False).arg == q
def _to_matrix(expr):
if expr in subst:
return subst[expr]
if isinstance(expr, Pow):
return MatPow(_to_matrix(expr.args[0]), expr.args[1])
elif isinstance(expr, (Add, Mul)):
return reduce(funcs[expr.func], [_to_matrix(a) for a in expr.args])
else:
return expr * Identity(x)
def test_doit_with_MatrixBase():
X = ImmutableMatrix([[1, 2], [3, 4]])
assert MatPow(X, 0).doit() == ImmutableMatrix(Identity(2))
assert MatPow(X, 1).doit() == X
assert MatPow(X, 2).doit() == X**2
assert MatPow(X, -1).doit() == X.inv()
assert MatPow(X, -2).doit() == (X.inv())**2
# less expensive than testing on a 2x2
assert MatPow(ImmutableMatrix([4]), S.Half).doit() == ImmutableMatrix([2])
X = ImmutableMatrix([[0, 2], [0, 4]]) # det() == 0
raises(ValueError, lambda: MatPow(X,-1).doit())
raises(ValueError, lambda: MatPow(X,-2).doit())
def test_identity_power():
k = Identity(n)
assert MatPow(k, 4).doit() == k
assert MatPow(k, n).doit() == k
assert MatPow(k, -3).doit() == k
assert MatPow(k, 0).doit() == k
l = Identity(3)
assert MatPow(l, n).doit() == l
assert MatPow(l, -1).doit() == l
assert MatPow(l, 0).doit() == l
def test_as_explicit():
A = ImmutableMatrix([[1, 2], [3, 4]])
assert MatPow(A, 0).as_explicit() == ImmutableMatrix(Identity(2))
assert MatPow(A, 1).as_explicit() == A
assert MatPow(A, 2).as_explicit() == A**2
assert MatPow(A, -1).as_explicit() == A.inv()
assert MatPow(A, -2).as_explicit() == (A.inv())**2
# less expensive than testing on a 2x2
A = ImmutableMatrix([4]);
assert MatPow(A, S.Half).as_explicit() == A**S.Half
def test_doit_nested_MatrixExpr():
X = ImmutableMatrix([[1, 2], [3, 4]])
Y = ImmutableMatrix([[2, 3], [4, 5]])
assert MatPow(MatMul(X, Y), 2).doit() == (X*Y)**2
assert MatPow(MatAdd(X, Y), 2).doit() == (X + Y)**2
def test_doit_symbol():
assert MatPow(C, 0).doit() == Identity(n)
assert MatPow(C, 1).doit() == C
assert MatPow(C, -1).doit() == C.I
for r in [2, S.Half, S.Pi, n]:
assert MatPow(C, r).doit() == MatPow(C, r)
def test_doit_square_MatrixSymbol_symsize():
assert MatPow(C, 0).doit() == Identity(n)
assert MatPow(C, 1).doit() == C
for r in [2, -1, pi]:
assert MatPow(C, r).doit() == MatPow(C, r)
def test_doit_nonsquare_MatrixSymbol():
assert MatPow(A, 1).doit() == A
for r in [0, 2, -1, pi]:
assert MatPow(A, r).doit() == MatPow(A, r)
def test_nonsquare():
A = MatrixSymbol('A', 2, 3)
B = ImmutableMatrix([[1, 2, 3], [4, 5, 6]])
for r in [-1, 0, 1, 2, S.Half, S.Pi, n]:
raises(NonSquareMatrixError, lambda: MatPow(A, r))
raises(NonSquareMatrixError, lambda: MatPow(B, r))
def test_as_explicit_nonsquare_symbol():
X = MatrixSymbol('X', 2, 3)
assert MatPow(X, 1).as_explicit() == X.as_explicit()
for r in [0, 2, S.Half, S.Pi]:
raises(ShapeError, lambda: MatPow(X, r).as_explicit())
def test_as_explicit_symbol():
X = MatrixSymbol('X', 2, 2)
assert MatPow(X, 0).as_explicit() == ImmutableMatrix(Identity(2))
assert MatPow(X, 1).as_explicit() == X.as_explicit()
assert MatPow(X, 2).as_explicit() == (X.as_explicit())**2
def test_inverse_matpow_canonicalization():
A = MatrixSymbol('A', 3, 3)
assert Inverse(MatPow(A, 3)).doit() == MatPow(Inverse(A), 3).doit()
def test_doit_drills_down():
X = ImmutableMatrix([[1, 2], [3, 4]])
Y = ImmutableMatrix([[2, 3], [4, 5]])
assert MatMul(X, MatPow(Y, 2)).doit() == X * Y**2
assert MatMul(C, Transpose(D * C)).doit().args == (C, C.T, D.T)
def test_doit_equals_pow(): #17179
X = ImmutableMatrix ([[1,0],[0,1]])
assert MatPow(X, n).doit() == X**n == X
