from sympy.functions.elementary.miscellaneous import sqrt from sympy.simplify.powsimp import powsimp from sympy.testing.pytest import raises from sympy.core.expr import unchanged from sympy.core import symbols, S from sympy.matrices import Identity, MatrixSymbol, ImmutableMatrix, ZeroMatrix, OneMatrix, Matrix from sympy.matrices.common import NonSquareMatrixError from sympy.matrices.expressions import MatPow, MatAdd, MatMul from sympy.matrices.expressions.inverse import Inverse from sympy.matrices.expressions.matexpr import MatrixElement n, m, l, k = symbols('n m l k', integer=True) A = MatrixSymbol('A', n, m) B = MatrixSymbol('B', m, l) C = MatrixSymbol('C', n, n) D = MatrixSymbol('D', n, n) E = MatrixSymbol('E', m, n) 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_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_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_matrix(): 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_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_matrix(): 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_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_doit_equals_pow(): #17179 X = ImmutableMatrix ([[1,0],[0,1]]) assert MatPow(X, n).doit() == X**n == X 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_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_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_OneMatrix_power(): o = OneMatrix(3, 3) assert o ** 0 == Identity(3) assert o ** 1 == o assert o * o == o ** 2 == 3 * o assert o * o * o == o ** 3 == 9 * o o = OneMatrix(n, n) assert o * o == o ** 2 == n * o # powsimp necessary as n ** (n - 2) * n does not produce n ** (n - 1) assert powsimp(o ** (n - 1) * o) == o ** n == n ** (n - 1) * o def test_transpose_power(): from sympy.matrices.expressions.transpose import Transpose as TP assert (C*D).T**5 == ((C*D)**5).T == (D.T * C.T)**5 assert ((C*D).T**5).T == (C*D)**5 assert (C.T.I.T)**7 == C**-7 assert (C.T**l).T**k == C**(l*k) assert ((E.T * A.T)**5).T == (A*E)**5 assert ((A*E).T**5).T**7 == (A*E)**35 assert TP(TP(C**2 * D**3)**5).doit() == (C**2 * D**3)**5 assert ((D*C)**-5).T**-5 == ((D*C)**25).T assert (((D*C)**l).T**k).T == (D*C)**(l*k) 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_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_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_no_exponentiation(): # if this passes, Pow.as_numer_denom should recognize # MatAdd as exponent raises(NotImplementedError, lambda: 3**(-2*C))