def __init__(self, A, b=0, parent=None, convert=True, check=True):
        r"""
        Create element of an affine group.

        TESTS::

            sage: G = AffineGroup(4, GF(5))
            sage: g = G.random_element()
            sage: TestSuite(g).run()
        """
        if is_Matrix(A) and A.nrows() == A.ncols() == parent.degree()+1:
            g = A
            A = g.submatrix(0,0,2,2)
            d = parent.degree()
            b = [ g[i,d] for i in range(d) ]
            convert = True
        if convert:
            A = parent.matrix_space()(A)
            b = parent.vector_space()(b)
        if check:
            # Note: the coercion framework expects that we raise TypeError for invalid input
            if not is_Matrix(A):
                raise TypeError('A must be a matrix')
            if not (A.parent() is parent.matrix_space()):
                raise TypeError('A must be an element of '+str(parent.matrix_space()))
            if not (b.parent() is parent.vector_space()):
                raise TypeError('b must be an element of '+str(parent.vector_space()))
            parent._element_constructor_check(A, b)
        super(AffineGroupElement, self).__init__(parent)
        self._A = A
        self._b = b

    def __classcall_private__(self, arg0, arg1=None, names=None):
        """
        Choose the correct parent based upon input.

        TESTS:

        We check arguments with passing in an associative algebra::

            sage: cat = Algebras(QQ).WithBasis().FiniteDimensional()
            sage: C = CombinatorialFreeModule(QQ, ['x','y','z'], category=cat)
            sage: J1 = JordanAlgebra(C, names=['a','b','c'])
            sage: J2.<a,b,c> = JordanAlgebra(C)
            sage: J1 is J2
            True

        We check with passing in a symmetric bilinear form::

            sage: m = matrix([[0,1],[1,1]])
            sage: J1 = JordanAlgebra(m)
            sage: J2 = JordanAlgebra(QQ, m)
            sage: J3 = JordanAlgebra(m, QQ)
            sage: J1 is J2
            False
            sage: J2 is J3
            True
            sage: J4 = JordanAlgebra(ZZ, m)
            sage: J1 is J4
            True
            sage: m = matrix(QQ, [[0,1],[1,1]])
            sage: J1 = JordanAlgebra(m)
            sage: J1 is J2
            True
        """
        if names is not None:
            if isinstance(names, str):
                names = names.split(',')
            names = tuple(names)

        if arg1 is None:
            if not is_Matrix(arg0):
                if arg0.base_ring().characteristic() == 2:
                    raise ValueError("the base ring cannot have characteristic 2")
                return SpecialJordanAlgebra(arg0, names)
            arg0, arg1 = arg0.base_ring(), arg0
        elif is_Matrix(arg0):
            arg0, arg1 = arg1, arg0

        # arg0 is the base ring and arg1 is a matrix
        if not arg1.is_symmetric():
            raise ValueError("the bilinear form is not symmetric")

        arg1 = arg1.change_ring(arg0) # This makes a copy
        arg1.set_immutable()
        return JordanAlgebraSymmetricBilinear(arg0, arg1, names=names)

    def _acted_upon_(self, g, self_on_left):
        r"""
        Implements the left action of `SL_2(\ZZ)` on self.

        EXAMPLES::

            sage: g = matrix(ZZ, 2, [1,1,0,1]); g
            [1 1]
            [0 1]
            sage: g * Cusp(2,5)
            7/5
            sage: Cusp(2,5) * g
            Traceback (most recent call last):
            ...
            TypeError: unsupported operand parent(s) for '*': 'Set P^1(QQ) of all cusps' and 'Full MatrixSpace of 2 by 2 dense matrices over Integer Ring'
            sage: h = matrix(ZZ, 2, [12,3,-100,7])
            sage: h * Cusp(2,5)
            -13/55
            sage: Cusp(2,5)._acted_upon_(h, False)
            -13/55
            sage: (h*g) * Cusp(3,7) == h * (g * Cusp(3,7))
            True

            sage: cm = sage.structure.element.get_coercion_model()
            sage: cm.explain(MatrixSpace(ZZ, 2), Cusps)
            Action discovered.
                Left action by Full MatrixSpace of 2 by 2 dense matrices over Integer Ring on Set P^1(QQ) of all cusps
            Result lives in Set P^1(QQ) of all cusps
            Set P^1(QQ) of all cusps
        """
        if not self_on_left:
            if (is_Matrix(g) and g.base_ring() is ZZ
                    and g.ncols() == 2 and g.nrows() == 2):
                a, b, c, d = g.list()
                return Cusp(a*self.__a + b*self.__b, c*self.__a + d*self.__b)

    def __call__(self, f, check=True, unitary=True):
        """
        Construct a homomorphism.

        .. TODO::

            Implement taking generator images and converting them to a matrix.

        EXAMPLES::

            sage: A = FiniteDimensionalAlgebra(QQ, [Matrix([1])])
            sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
            sage: H = Hom(A, B)
            sage: H(Matrix([[1, 0]]))
            Morphism from Finite-dimensional algebra of degree 1 over Rational Field to
             Finite-dimensional algebra of degree 2 over Rational Field given by matrix
            [1 0]
        """
        if isinstance(f, FiniteDimensionalAlgebraMorphism):
            if f.parent() is self:
                return f
            if f.parent() == self:
                return FiniteDimensionalAlgebraMorphism(self, f._matrix, check, unitary)
        elif is_Matrix(f):
            return FiniteDimensionalAlgebraMorphism(self, f, check, unitary)
        try:
            from sage.matrix.constructor import Matrix
            return FiniteDimensionalAlgebraMorphism(self, Matrix(f), check, unitary)
        except Exception:
            return RingHomset_generic.__call__(self, f, check)

def print_obj(out_stream, obj):
    """
    Print an object. This function is used internally by the displayhook.

    EXAMPLES::

        sage: import sage.misc.displayhook, sys

    For most objects, printing is done simply using their repr::

        sage: sage.misc.displayhook.print_obj(sys.stdout, 'Hello, world!')
        'Hello, world!'
        sage: sage.misc.displayhook.print_obj(sys.stdout, (1, 2, 3, 4))
        (1, 2, 3, 4)

    We demonstrate the special format for lists of matrices::

        sage: sage.misc.displayhook.print_obj(sys.stdout, \
                [matrix([[1], [2]]), matrix([[3], [4]])])
        [
        [1]  [3]
        [2], [4]
        ]
    """
    # We only apply the special formatting to lists (or tuples) where the first
    # element is a matrix. This should cover most cases.
    if isinstance(obj, (tuple, list)):
        if len(obj) > 0 and is_Matrix(obj[0]):
            if _check_tall_list_and_print(out_stream, obj):
                return
    print >>out_stream, `obj`

    def __call__(self, x):
        """
        EXAMPLES::
        
            sage: W = WeylGroup(['A',2])
            sage: W(1)
            [1 0 0]
            [0 1 0]
            [0 0 1]
        
        ::
        
            sage: W(2)
            Traceback (most recent call last):
            ...
            TypeError: no way to coerce element into self.

            sage: W2 = WeylGroup(['A',3])
            sage: W(1) in W2  # indirect doctest
            False

        """
        if isinstance(x, self.element_class) and x.parent() is self:
            return x
        from sage.matrix.matrix import is_Matrix
        if not (x in ZZ or is_Matrix(x)): # this should be handled by self.matrix_space()(x)
            raise TypeError, "no way to coerce element into self"
        M = self.matrix_space()(x)
        # This is really bad, especially for infinite groups!
        # TODO: compute the image of rho, compose by s_i until back in
        # the fundamental chamber. Return True iff the matrix is the identity
        g = self._element_constructor_(M)
        if not gap(g) in gap(self):
            raise TypeError, "no way to coerce element into self."
        return g

    def _is_even_symmetric_matrix_(self, A, R=None):
        """
        Tests if a matrix is symmetric, defined over R, and has even diagonal in R.

        INPUT:
            A -- matrix

            R -- ring

        EXAMPLES::

            sage: Q = QuadraticForm(ZZ, 2, [2,3,5])
            sage: A = Q.matrix()
            sage: A
            [ 4  3]
            [ 3 10]
            sage: Q._is_even_symmetric_matrix_(A)
            True
            sage: A[0,0] = 1
            sage: Q._is_even_symmetric_matrix_(A)
            False

        """
        if not is_Matrix(A):
            raise TypeError("A is not a matrix.")

        ring_coerce_test = True
        if R == None:            ## This allows us to omit the ring from the variables, and take it from the matrix
            R = A.base_ring()
            ring_coerce_test = False

        if not isinstance(R, Ring):
            raise TypeError("R is not a ring.")

        if not A.is_square():
            return False

        ## Test that the matrix is symmetric
        n = A.nrows()
        for i in range(n):
            for j in range(i+1, n):
                if A[i,j] != A[j,i]:
                    return False

        ## Test that all entries coerce to R
        if not ((A.base_ring() == R) or (ring_coerce_test == True)):
            try:
                for i in range(n):
                    for j in range(i, n):
                        x = R(A[i,j])
            except Exception:
                return False

        ## Test that the diagonal is even (if 1/2 isn't in R)
        if not R(2).is_unit():
            for i in range(n):
                if not is_even(R(A[i,i])):
                    return False

        return True

    def __init__(self, A, gens=None, given_by_matrix=False):
        """
        EXAMPLES::

            sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
            sage: I = A.ideal(A([0,1]))
            sage: TestSuite(I).run(skip="_test_category") # Currently ideals are not using the category framework
        """
        k = A.base_ring()
        n = A.degree()
        if given_by_matrix:
            self._basis_matrix = gens
            gens = gens.rows()
        elif gens is None:
            self._basis_matrix = Matrix(k, 0, n)
        elif isinstance(gens, (list, tuple)):
            B = [FiniteDimensionalAlgebraIdeal(A, x).basis_matrix() for x in gens]
            B = reduce(lambda x, y: x.stack(y), B, Matrix(k, 0, n))
            self._basis_matrix = B.echelon_form().image().basis_matrix()
        elif is_Matrix(gens):
            gens = FiniteDimensionalAlgebraElement(A, gens)
        elif isinstance(gens, FiniteDimensionalAlgebraElement):
            gens = gens.vector()
            B = Matrix([gens * b for b in A.table()])
            self._basis_matrix = B.echelon_form().image().basis_matrix()
        Ideal_generic.__init__(self, A, gens)

    def __rmul__(self, other):
        r"""
        Implement the action of matrices on points of hyperbolic space.

        EXAMPLES::

            sage: A = matrix(2, [0, 1, 1, 0])
            sage: A = HyperbolicPlane().UHP().get_isometry(A)
            sage: A * HyperbolicPlane().UHP().get_point(2 + I)
            Point in UHP 1/5*I + 2/5

        We also lift matrices into isometries::

            sage: B = diagonal_matrix([-1, -1, 1])
            sage: B = HyperbolicPlane().HM().get_isometry(B)
            sage: B * HyperbolicPlane().HM().get_point((0, 1, sqrt(2)))
            Point in HM (0, -1, sqrt(2))
        """
        if isinstance(other, HyperbolicIsometry):
            return other(self)
        elif is_Matrix(other):
            # TODO: Currently the __mul__ from the matrices gets called first
            #    and returns an error instead of calling this method
            A = self.parent().get_isometry(other)
            return A(self)
        else:
            raise TypeError("unsupported operand type(s) for *:"
                            "{0} and {1}".format(self, other))

def normalize_square_matrices(matrices):
    """
    Find a common space for all matrices.

    OUTPUT:

    A list of matrices, all elements of the same matrix space.

    EXAMPLES::

        sage: from sage.groups.matrix_gps.finitely_generated import normalize_square_matrices
        sage: m1 = [[1,2],[3,4]]
        sage: m2 = [2, 3, 4, 5]
        sage: m3 = matrix(QQ, [[1/2,1/3],[1/4,1/5]])
        sage: m4 = MatrixGroup(m3).gen(0)
        sage: normalize_square_matrices([m1, m2, m3, m4])
        [
        [1 2]  [2 3]  [1/2 1/3]  [1/2 1/3]
        [3 4], [4 5], [1/4 1/5], [1/4 1/5]
        ]
    """
    deg = []
    gens = []
    for m in matrices:
        if is_MatrixGroupElement(m):
            deg.append(m.parent().degree())
            gens.append(m.matrix())
            continue
        if is_Matrix(m):
            if not m.is_square():
                raise TypeError('matrix must be square')
            deg.append(m.ncols())
            gens.append(m)
            continue
        try:
            m = list(m)
        except TypeError:
            gens.append(m)
            continue
        if isinstance(m[0], (list, tuple)):
            m = [list(_) for _ in m]
            degree = ZZ(len(m))
        else:
            degree, rem = ZZ(len(m)).sqrtrem()
            if rem!=0:
                raise ValueError('list of plain numbers must have square integer length')
        deg.append(degree)
        gens.append(matrix(degree, degree, m))
    deg = set(deg)
    if len(set(deg)) != 1:
        raise ValueError('not all matrices have the same size')
    gens = Sequence(gens, immutable=True)
    MS = gens.universe()
    if not is_MatrixSpace(MS):
        raise TypeError('all generators must be matrices')
    if MS.nrows() != MS.ncols():
        raise ValueError('matrices must be square')
    return gens

def from_incidence_matrix(G, M, loops=False, multiedges=False, weighted=False):
    r"""
    Fill ``G`` with the data of an incidence matrix.

    INPUT:

    - ``G`` -- a graph

    - ``M`` -- an incidence matrix

    - ``loops``, ``multiedges``, ``weighted`` (booleans) -- whether to consider
      the graph as having loops, multiple edges, or weights. Set to ``False`` by default.

    EXAMPLE::

        sage: from sage.graphs.graph_input import from_incidence_matrix
        sage: g = Graph()
        sage: from_incidence_matrix(g, graphs.PetersenGraph().incidence_matrix())
        sage: g.is_isomorphic(graphs.PetersenGraph())
        True
    """
    from sage.matrix.matrix import is_Matrix
    assert is_Matrix(M)

    oriented = any(M[pos] < 0 for pos in M.nonzero_positions(copy=False))

    positions = []
    for i in range(M.ncols()):
        NZ = M.nonzero_positions_in_column(i)
        if len(NZ) == 1:
            if oriented:
                raise ValueError("Column {} of the (oriented) incidence "
                                 "matrix contains only one nonzero value".format(i))
            elif M[NZ[0],i] != 2:
                raise ValueError("Each column of a non-oriented incidence "
                                 "matrix must sum to 2, but column {} does not".format(i))
            if loops is None:
                loops = True
            positions.append((NZ[0],NZ[0]))
        elif len(NZ) != 2 or \
             (oriented and not ((M[NZ[0],i] == +1 and M[NZ[1],i] == -1) or \
                                (M[NZ[0],i] == -1 and M[NZ[1],i] == +1))) or \
             (not oriented and (M[NZ[0],i] != 1 or M[NZ[1],i] != 1)):
            msg  = "There must be one or two nonzero entries per column in an incidence matrix. "
            msg += "Got entries {} in column {}".format([M[j,i] for j in NZ], i)
            raise ValueError(msg)
        else:
            positions.append(tuple(NZ))

    if weighted   is None: G._weighted  = False
    if multiedges is None:
        total = len(positions)
        multiedges = (len(set(positions)) < total  )
    G.allow_loops(False if loops is None else loops, check=False)
    G.allow_multiple_edges(multiedges, check=False)
    G.add_vertices(range(M.nrows()))
    G.add_edges(positions)

    def __init__(self, A, elt=None, check=True):
        """
        TESTS::

            sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1,0], [0,1]]), Matrix([[0,1], [0,0]])])
            sage: A(QQ(4))
            Traceback (most recent call last):
            ...
            TypeError: elt should be a vector, a matrix, or an element of the base field

            sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]), Matrix([[0,1], [-1,0]])])
            sage: elt = B(Matrix([[1,1], [-1,1]])); elt
            e0 + e1
            sage: TestSuite(elt).run()
            sage: B(Matrix([[0,1], [1,0]]))
            Traceback (most recent call last):
            ...
            ValueError: matrix does not define an element of the algebra
        """
        AlgebraElement.__init__(self, A)
        k = A.base_ring()
        n = A.degree()
        if elt is None:
            self._vector = vector(k, n)
            self._matrix = Matrix(k, n)
        else:
            if isinstance(elt, int):
                elt = Integer(elt)
            elif isinstance(elt, list):
                elt = vector(elt)
            if A == elt.parent():
                self._vector = elt._vector.base_extend(k)
                self._matrix = elt._matrix.base_extend(k)
            elif k.has_coerce_map_from(elt.parent()):
                e = k(elt)
                if e == 0:
                    self._vector = vector(k, n)
                    self._matrix = Matrix(k, n)
                elif A.is_unitary():
                    self._vector = A._one * e
                    self._matrix = Matrix.identity(k, n) * e
                else:
                    raise TypeError("algebra is not unitary")
            elif is_Vector(elt):
                self._vector = elt.base_extend(k)
                self._matrix = Matrix(k, sum([elt[i] * A.table()[i] for i in xrange(n)]))
            elif is_Matrix(elt):
                if not A.is_unitary():
                    raise TypeError("algebra is not unitary")
                self._vector = A._one * elt
                if not check or sum([self._vector[i]*A.table()[i] for i in xrange(n)]) == elt:
                    self._matrix = elt
                else:
                    raise ValueError("matrix does not define an element of the algebra")
            else:
                raise TypeError("elt should be a vector, a matrix, " +
                                "or an element of the base field")

def from_oriented_incidence_matrix(G, M, loops=False, multiedges=False, weighted=False):
    r"""
    Fill ``G`` with the data of an *oriented* incidence matrix.

    An oriented incidence matrix is the incidence matrix of a directed graph, in
    which each non-loop edge corresponds to a `+1` and a `-1`, indicating its
    source and destination.

    INPUT:

    - ``G`` -- a :class:`DiGraph`

    - ``M`` -- an incidence matrix

    - ``loops``, ``multiedges``, ``weighted`` (booleans) -- whether to consider
      the graph as having loops, multiple edges, or weights. Set to ``False`` by default.

    EXAMPLES::

        sage: from sage.graphs.graph_input import from_oriented_incidence_matrix
        sage: g = DiGraph()
        sage: from_oriented_incidence_matrix(g, digraphs.Circuit(10).incidence_matrix())
        sage: g.is_isomorphic(digraphs.Circuit(10))
        True

    TESTS:

    Fix bug reported in :trac:`22985`::

        sage: DiGraph(matrix ([[1,0,0,1],[0,0,1,1],[0,0,1,1]]).transpose())
        Traceback (most recent call last):
        ...
        ValueError: each column represents an edge: -1 goes to 1
    """
    from sage.matrix.matrix import is_Matrix
    assert is_Matrix(M)

    positions = []
    for c in M.columns():
        NZ = c.nonzero_positions()
        if len(NZ) != 2:
            raise ValueError("There must be two nonzero entries (-1 & 1) per column.")
        L = sorted(set(c.list()))
        if L != [-1,0,1]:
            raise ValueError("each column represents an edge: -1 goes to 1")
        if c[NZ[0]] == -1:
            positions.append(tuple(NZ))
        else:
            positions.append((NZ[1],NZ[0]))
    if weighted   is None: weighted  = False
    if multiedges is None:
        total = len(positions)
        multiedges = (  len(set(positions)) < total  )
    G.allow_loops(True if loops else False,check=False)
    G.allow_multiple_edges(multiedges,check=False)
    G.add_vertices(range(M.nrows()))
    G.add_edges(positions)

def from_seidel_adjacency_matrix(G, M):
    r"""
    Fill ``G`` with the data of a Seidel adjacency matrix.

    INPUT:

    - ``G`` -- a graph

    - ``M`` -- a Seidel adjacency matrix

    EXAMPLE::

        sage: from sage.graphs.graph_input import from_seidel_adjacency_matrix
        sage: g = Graph()
        sage: from_seidel_adjacency_matrix(g, graphs.PetersenGraph().seidel_adjacency_matrix())
        sage: g.is_isomorphic(graphs.PetersenGraph())
        True
    """
    from sage.matrix.matrix import is_Matrix
    from sage.rings.integer_ring import ZZ
    assert is_Matrix(M)

    if M.base_ring() != ZZ:
        try:
            M = M.change_ring(ZZ)
        except TypeError:
            raise ValueError("Graph's Seidel adjacency matrix must"+
                             " have only 0,1,-1 integer entries")

    if M.is_sparse():
        entries = set(M[i,j] for i,j in M.nonzero_positions())
    else:
        entries = set(M.list())

    if any(e <  -1 or e > 1 for e in entries):
        raise ValueError("Graph's Seidel adjacency matrix must"+
                         " have only 0,1,-1 integer entries")
    if any(i==j for i,j in M.nonzero_positions()):
        raise ValueError("Graph's Seidel adjacency matrix must"+
                         " have 0s on the main diagonal")
    if not M.is_symmetric():
        raise ValueError("Graph's Seidel adjacency matrix must"+
                         " be symmetric")
    G.add_vertices(range(M.nrows()))
    e = []
    for i,j in M.nonzero_positions():
       if i <= j and M[i,j] < 0:
                e.append((i,j))
    G.add_edges(e)

    def __init__(self, k, table, names='e', assume_associative=False, category=None):
        """
        TESTS::

            sage: A = FiniteDimensionalAlgebra(QQ, [])
            sage: A
            Finite-dimensional algebra of degree 0 over Rational Field
            sage: type(A)
            <class 'sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra.FiniteDimensionalAlgebra_with_category'>
            sage: TestSuite(A).run()

            sage: B = FiniteDimensionalAlgebra(GF(7), [Matrix([1])])
            sage: B
            Finite-dimensional algebra of degree 1 over Finite Field of size 7
            sage: TestSuite(B).run()

            sage: C = FiniteDimensionalAlgebra(CC, [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
            sage: C
            Finite-dimensional algebra of degree 2 over Complex Field with 53 bits of precision
            sage: TestSuite(C).run()

            sage: FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]])])
            Traceback (most recent call last):
            ...
            ValueError: input is not a multiplication table

            sage: D.<a,b> = FiniteDimensionalAlgebra(RR, [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [-1, 0]])])
            sage: D.gens()
            (a, b)

            sage: E = FiniteDimensionalAlgebra(QQ, [Matrix([0])])
            sage: E.gens()
            (e,)
        """
        n = len(table)
        self._table = [b.base_extend(k) for b in table]
        if not all([is_Matrix(b) and b.dimensions() == (n, n) for b in table]):
            raise ValueError("input is not a multiplication table")
        self._assume_associative = assume_associative
        # No further validity checks necessary!
        if category is None:
            category = FiniteDimensionalAlgebrasWithBasis(k)
        Algebra.__init__(self, base_ring=k, names=names, category=category)

    def __init__(self, parent, M, check=True, convert=True):
        r"""
        Element of a matrix group over a generic ring.

        The group elements are implemented as Sage matrices.

        INPUT:

        - ``M`` -- a matrix.

        - ``parent`` -- the parent.

        - ``check`` -- bool (default: ``True``). If true does some
          type checking.

        - ``convert`` -- bool (default: ``True``). If true convert
          ``M`` to the right matrix space.

        TESTS::

            sage: MS = MatrixSpace(GF(3),2,2)
            sage: G = MatrixGroup(MS([[1,0],[0,1]]), MS([[1,1],[0,1]]))
            sage: G.gen(0)
            [1 0]
            [0 1]
            sage: g = G.random_element()
            sage: TestSuite(g).run()
        """
        if isinstance(M, GapElement):
            ElementLibGAP.__init__(self, parent, M)
            return
        if convert:
            M = parent.matrix_space()(M)
        from sage.libs.gap.libgap import libgap

        M_gap = libgap(M)
        if check:
            if not is_Matrix(M):
                raise TypeError("M must be a matrix")
            if M.parent() is not parent.matrix_space():
                raise TypeError("M must be a in the matrix space of the group")
            parent._check_matrix(M, M_gap)
        ElementLibGAP.__init__(self, parent, M_gap)

    def __init__(self, parent, A):
        r"""
        Initialise an element from a matrix. This *must* be over the base ring
        of self and have the right size.

        This is a bit overkill as similar checks will be performed by the call
        and coerce methods of the parent of self, but it can't hurt to be
        paranoid. Any fancy coercion / base_extension / etc happens there, not
        here.

        TESTS::

            sage: T = ModularForms(Gamma0(7), 4).hecke_algebra()
            sage: M = sage.modular.hecke.hecke_operator.HeckeAlgebraElement_matrix(T, matrix(QQ,3,[2,3,0,1,2,3,7,8,9])); M
            Hecke operator on Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(7) of weight 4 over Rational Field defined by:
            [2 3 0]
            [1 2 3]
            [7 8 9]
            sage: loads(dumps(M)) == M
            True
            sage: sage.modular.hecke.hecke_operator.HeckeAlgebraElement_matrix(T, matrix(Integers(2),3,[2,3,0,1,2,3,7,8,9]))
            Traceback (most recent call last):
            ...
            TypeError: base ring of matrix (Ring of integers modulo 2) does not match base ring of space (Rational Field)
            sage: sage.modular.hecke.hecke_operator.HeckeAlgebraElement_matrix(T, matrix(QQ,2,[2,3,0,1]))
            Traceback (most recent call last):
            ...
            TypeError: A must be a square matrix of rank 3
        """
        HeckeAlgebraElement.__init__(self, parent)
        from sage.matrix.matrix import is_Matrix

        if not is_Matrix(A):
            raise TypeError("A must be a matrix")
        if not A.base_ring() == self.parent().base_ring():
            raise TypeError, "base ring of matrix (%s) does not match base ring of space (%s)" % (
                A.base_ring(),
                self.parent().base_ring(),
            )
        if not A.nrows() == A.ncols() == self.parent().module().rank():
            raise TypeError, "A must be a square matrix of rank %s" % self.parent().module().rank()
        self.__matrix = A

def jacobian(functions, variables):
    """
    Return the Jacobian matrix, which is the matrix of partial
    derivatives in which the i,j entry of the Jacobian matrix is the
    partial derivative diff(functions[i], variables[j]).

    EXAMPLES::

        sage: x,y = var('x,y')
        sage: g=x^2-2*x*y
        sage: jacobian(g, (x,y))
        [2*x - 2*y      -2*x]

    The Jacobian of the Jacobian should give us the "second derivative", which is the Hessian matrix::

        sage: jacobian(jacobian(g, (x,y)), (x,y))
        [ 2 -2]
        [-2  0]
        sage: g.hessian()
        [ 2 -2]
        [-2  0]

        sage: f=(x^3*sin(y), cos(x)*sin(y), exp(x))
        sage: jacobian(f, (x,y))
        [  3*x^2*sin(y)     x^3*cos(y)]
        [-sin(x)*sin(y)  cos(x)*cos(y)]
        [           e^x              0]
        sage: jacobian(f, (y,x))
        [    x^3*cos(y)   3*x^2*sin(y)]
        [ cos(x)*cos(y) -sin(x)*sin(y)]
        [             0            e^x]

    """
    if is_Matrix(functions) and (functions.nrows()==1 or functions.ncols()==1):
        functions = functions.list()
    elif not (isinstance(functions, (tuple, list)) or is_Vector(functions)):
        functions = [functions]

    if not isinstance(variables, (tuple, list)) and not is_Vector(variables):
        variables = [variables]

    return matrix([[diff(f, v) for v in variables] for f in functions])

def is_generalized_cartan_matrix(M):
    """
    Return ``True`` if ``M`` is a generalized Cartan matrix. For a definition
    of a generalized Cartan matrix, see :class:`CartanMatrix`.

    EXAMPLES::

        sage: from sage.combinat.root_system.cartan_matrix import is_generalized_cartan_matrix
        sage: M = matrix([[2,-1,-2], [-1,2,-1], [-2,-1,2]])
        sage: is_generalized_cartan_matrix(M)
        True
        sage: M = matrix([[2,-1,-2], [-1,2,-1], [0,-1,2]])
        sage: is_generalized_cartan_matrix(M)
        False
        sage: M = matrix([[1,-1,-2], [-1,2,-1], [-2,-1,2]])
        sage: is_generalized_cartan_matrix(M)
        False

    A non-symmetrizable example::

        sage: M = matrix([[2,-1,-2], [-1,2,-1], [-1,-1,2]])
        sage: is_generalized_cartan_matrix(M)
        True
    """
    if not is_Matrix(M):
        return False
    if not M.is_square():
        return False
    n = M.ncols()
    for i in xrange(n):
        if M[i,i] != 2:
            return False
        for j in xrange(i+1, n):
            if M[i,j] > 0 or M[j,i] > 0:
                return False
            elif M[i,j] == 0 and M[j,i] != 0:
                return False
            elif M[j,i] == 0 and M[i,j] != 0:
                return False
    return True

    def write_matrix(self, mat, filename):
        r"""
        Write the matrix ``mat`` to the file ``filename`` in 4ti2 format.

        INPUT:

        - ``mat`` - A matrix of integers or something that can be
           converted to that.
        - ``filename`` - A file name not including a path.

        EXAMPLES::

            sage: from sage.interfaces.four_ti_2 import four_ti_2
            sage: four_ti_2.write_matrix([[1,2],[3,4]], "test_file")
        """
        from sage.matrix.constructor import matrix
        from sage.matrix.matrix import is_Matrix
        if not is_Matrix(mat):
            mat = matrix(ZZ, mat)
        if mat.base_ring() != ZZ:
            mat = mat.change_ring(ZZ)

        self.write_array(mat, mat.nrows(), mat.ncols(), filename)