def test_BlockMatrix():
    A = MatrixSymbol('A', n, m)
    B = MatrixSymbol('B', n, k)
    C = MatrixSymbol('C', l, m)
    D = MatrixSymbol('D', l, k)
    M = MatrixSymbol('M', m + k, p)
    N = MatrixSymbol('N', l + n, k + m)
    X = BlockMatrix(Matrix([[A, B], [C, D]]))

    assert X.__class__(*X.args) == X

    # block_collapse does nothing on normal inputs
    E = MatrixSymbol('E', n, m)
    assert block_collapse(A + 2*E) == A + 2*E
    F = MatrixSymbol('F', m, m)
    assert block_collapse(E.T*A*F) == E.T*A*F

    assert X.shape == (l + n, k + m)
    assert X.blockshape == (2, 2)
    assert transpose(X) == BlockMatrix(Matrix([[A.T, C.T], [B.T, D.T]]))
    assert transpose(X).shape == X.shape[::-1]

    # Test that BlockMatrices and MatrixSymbols can still mix
    assert (X*M).is_MatMul
    assert X._blockmul(M).is_MatMul
    assert (X*M).shape == (n + l, p)
    assert (X + N).is_MatAdd
    assert X._blockadd(N).is_MatAdd
    assert (X + N).shape == X.shape

    E = MatrixSymbol('E', m, 1)
    F = MatrixSymbol('F', k, 1)

    Y = BlockMatrix(Matrix([[E], [F]]))

    assert (X*Y).shape == (l + n, 1)
    assert block_collapse(X*Y).blocks[0, 0] == A*E + B*F
    assert block_collapse(X*Y).blocks[1, 0] == C*E + D*F

    # block_collapse passes down into container objects, transposes, and inverse
    assert block_collapse(transpose(X*Y)) == transpose(block_collapse(X*Y))
    assert block_collapse(Tuple(X*Y, 2*X)) == (
        block_collapse(X*Y), block_collapse(2*X))

    # Make sure that MatrixSymbols will enter 1x1 BlockMatrix if it simplifies
    Ab = BlockMatrix([[A]])
    Z = MatrixSymbol('Z', *A.shape)
    assert block_collapse(Ab + Z) == A + Z

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 _eval_transpose(self):
     # Flip all the individual matrices
     matrices = [transpose(matrix) for matrix in self.blocks]
     # Make a copy
     M = Matrix(self.blockshape[0], self.blockshape[1], matrices)
     # Transpose the block structure
     M = M.transpose()
     return BlockMatrix(M)

def test_Identity():
    A = MatrixSymbol('A', n, m)
    In = Identity(n)
    Im = Identity(m)

    assert A*Im == A
    assert In*A == A

    assert transpose(In) == In
    assert In.inverse() == In
    assert In.conjugate() == In

def test_transpose():
    assert transpose(A*B) == Transpose(B)*Transpose(A)
    assert transpose(2*A*B) == 2*Transpose(B)*Transpose(A)
    assert transpose(2*I*C) == 2*I*Transpose(C)

    M = Matrix(2, 2, [1, 2 + I, 3, 4])
    MT = Matrix(2, 2, [1, 3, 2 + I, 4])
    assert transpose(M) == MT
    assert transpose(2*M) == 2*MT
    assert transpose(x*M) == x*MT
    assert transpose(MatMul(2, M)) == MatMul(2, MT).doit()

def test_transpose():
    Sq = MatrixSymbol('Sq', n, n)

    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_Identity():
    A = MatrixSymbol('A', n, m)
    i, j = symbols('i j')

    In = Identity(n)
    Im = Identity(m)

    assert A*Im == A
    assert In*A == A

    assert transpose(In) == In
    assert In.inverse() == In
    assert In.conjugate() == In

    assert In[i, j] != 0
    assert Sum(In[i, j], (i, 0, n-1), (j, 0, n-1)).subs(n,3).doit() == 3
    assert Sum(Sum(In[i, j], (i, 0, n-1)), (j, 0, n-1)).subs(n,3).doit() == 3

def test_ZeroMatrix():
    A = MatrixSymbol('A', n, m)
    Z = ZeroMatrix(n, m)

    assert A + Z == A
    assert A*Z.T == ZeroMatrix(n, n)
    assert Z*A.T == ZeroMatrix(n, n)
    assert A - A == ZeroMatrix(*A.shape)

    assert transpose(Z) == ZeroMatrix(m, n)
    assert Z.conjugate() == Z

    assert ZeroMatrix(n, n)**0 == Identity(n)
    with raises(ShapeError):
        Z**0
    with raises(ShapeError):
        Z**2

def test_OneMatrix():
    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 transpose(U) == OneMatrix(m, n)
    assert U.conjugate() == U

    assert OneMatrix(n, n) ** 0 == Identity(n)
    with raises(ShapeError):
        U ** 0
    with raises(ShapeError):
        U ** 2

    U = OneMatrix(n, n)
    assert U[1, 2] == 1

    U = OneMatrix(2, 3)
    assert U.as_explicit() == ImmutableMatrix.ones(2, 3)

 def _eval_transpose(self):
     return MatMul(*[transpose(arg) for arg in self.args[::-1]]).doit()

 def transpose(self):
     return transpose(self)

 def doit(self, **hints):
     arg = self.arg
     if hints.get('deep', True) and isinstance(arg, Basic):
         return transpose(arg.doit(**hints))
     else:
         return transpose(self.arg)

 def _eval_transpose(self):
     return MatAdd(*[transpose(arg) for arg in self.args]).doit()

 def _eval_conjugate(self):
     return transpose(self.arg)