r"""
Morphisms defined by a matrix

A matrix morphism is a morphism that is defined by multiplication
by a matrix. Elements of domain must either have a method
``vector()`` that returns a vector that the defining
matrix can hit from the left, or be coercible into vector space of
appropriate dimension.

EXAMPLES::

    sage: from sage.modules.matrix_morphism import MatrixMorphism, is_MatrixMorphism
    sage: V = QQ^3
    sage: T = End(V)
    sage: M = MatrixSpace(QQ,3)
    sage: I = M.identity_matrix()
    sage: m = MatrixMorphism(T, I); m
    Morphism defined by the matrix
    [1 0 0]
    [0 1 0]
    [0 0 1]
    sage: is_MatrixMorphism(m)
    True
    sage: m.charpoly('x')
    x^3 - 3*x^2 + 3*x - 1
    sage: m.base_ring()
    Rational Field
    sage: m.det()
    1
    sage: m.fcp('x')
    (x - 1)^3
    sage: m.matrix()
    [1 0 0]
    [0 1 0]
    [0 0 1]
    sage: m.rank()
    3
    sage: m.trace()
    3

AUTHOR:

- William Stein: initial versions

- David Joyner (2005-12-17): added examples

- William Stein (2005-01-07): added __reduce__

- Craig Citro (2008-03-18): refactored MatrixMorphism class

- Rob Beezer (2011-07-15): additional methods, bug fixes, documentation
"""

import sage.categories.morphism
import sage.categories.homset
import sage.matrix.all as matrix
from sage.structure.all import Sequence, parent
from sage.structure.richcmp import richcmp

def is_MatrixMorphism(x):
    """
    Return True if x is a Matrix morphism of free modules.

    EXAMPLES::

        sage: V = ZZ^2; phi = V.hom([3*V.0, 2*V.1])
        sage: sage.modules.matrix_morphism.is_MatrixMorphism(phi)
        True
        sage: sage.modules.matrix_morphism.is_MatrixMorphism(3)
        False
    """
    return isinstance(x, MatrixMorphism_abstract)

class MatrixMorphism_abstract(sage.categories.morphism.Morphism):
    def __init__(self, parent):
        """
        INPUT:


        -  ``parent`` - a homspace

        -  ``A`` - matrix


        EXAMPLES::

            sage: from sage.modules.matrix_morphism import MatrixMorphism
            sage: T = End(ZZ^3)
            sage: M = MatrixSpace(ZZ,3)
            sage: I = M.identity_matrix()
            sage: A = MatrixMorphism(T, I)
            sage: loads(A.dumps()) == A
            True
        """
        if not sage.categories.homset.is_Homset(parent):
            raise TypeError("parent must be a Hom space")
        sage.categories.morphism.Morphism.__init__(self, parent)

    def _richcmp_(self, other, op):
        """
        Compare two matrix morphisms.

        EXAMPLES::

            sage: V = ZZ^2
            sage: phi = V.hom([3*V.0, 2*V.1])
            sage: phi == 3
            False
            sage: phi == phi
            True
        """
        return richcmp(self.matrix(), other.matrix(), op)

    def _call_(self, x):
        """
        Evaluate this matrix morphism at an element of the domain.

        .. NOTE::

            Coercion is done in the generic :meth:`__call__` method,
            which calls this method.

        EXAMPLES::

            sage: V = QQ^3; W = QQ^2
            sage: H = Hom(V, W); H
            Set of Morphisms (Linear Transformations) from
            Vector space of dimension 3 over Rational Field to
            Vector space of dimension 2 over Rational Field
            sage: phi = H(matrix(QQ, 3, 2, range(6))); phi
            Vector space morphism represented by the matrix:
            [0 1]
            [2 3]
            [4 5]
            Domain: Vector space of dimension 3 over Rational Field
            Codomain: Vector space of dimension 2 over Rational Field
            sage: phi(V.0)
            (0, 1)
            sage: phi(V([1, 2, 3]))
            (16, 22)

        Last, we have a situation where coercion occurs::

            sage: U = V.span([[3,2,1]])
            sage: U.0
            (1, 2/3, 1/3)
            sage: phi(2*U.0)
            (16/3, 28/3)

        TESTS::

            sage: V = QQ^3; W = span([[1,2,3],[-1,2,5/3]], QQ)
            sage: phi = V.hom(matrix(QQ,3,[1..9]))

        We compute the image of some elements::

            sage: phi(V.0)    #indirect doctest
            (1, 2, 3)
            sage: phi(V.1)
            (4, 5, 6)
            sage: phi(V.0  - 1/4*V.1)
            (0, 3/4, 3/2)

        We restrict ``phi`` to ``W`` and compute the image of an element::

            sage: psi = phi.restrict_domain(W)
            sage: psi(W.0) == phi(W.0)
            True
            sage: psi(W.1) == phi(W.1)
            True
        """
        try:
            if parent(x) is not self.domain():
                x = self.domain()(x)
        except TypeError:
            raise TypeError("%s must be coercible into %s"%(x,self.domain()))
        if self.domain().is_ambient():
            x = x.element()
        else:
            x = self.domain().coordinate_vector(x)
        C = self.codomain()
        v = x.change_ring(C.base_ring()) * self.matrix()
        if not C.is_ambient():
            v = C.linear_combination_of_basis(v)
        # The call method of parents uses (coercion) morphisms.
        # Hence, in order to avoid recursion, we call the element
        # constructor directly; after all, we already know the
        # coordinates.
        return C._element_constructor_(v)

    def _call_with_args(self, x, args=(), kwds={}):
        """
        Like :meth:`_call_`, but takes optional and keyword arguments.

        EXAMPLES::

            sage: V = RR^2
            sage: f = V.hom(V.gens())
            sage: f._matrix *= I         # f is now invalid
            sage: f((1, 0))
            Traceback (most recent call last):
            ...
            TypeError: Unable to coerce entries (=[1.00000000000000*I, 0.000000000000000]) to coefficients in Real Field with 53 bits of precision
            sage: f((1, 0), coerce=False)
            (1.00000000000000*I, 0.000000000000000)

        """
        if self.domain().is_ambient():
            x = x.element()
        else:
            x = self.domain().coordinate_vector(x)
        C = self.codomain()
        v = x.change_ring(C.base_ring()) * self.matrix()
        if not C.is_ambient():
            v = C.linear_combination_of_basis(v)
        # The call method of parents uses (coercion) morphisms.
        # Hence, in order to avoid recursion, we call the element
        # constructor directly; after all, we already know the
        # coordinates.
        return C._element_constructor_(v, *args, **kwds)

    def __invert__(self):
        """
        Invert this matrix morphism.

        EXAMPLES::

            sage: V = QQ^2; phi = V.hom([3*V.0, 2*V.1])
            sage: phi^(-1)
            Vector space morphism represented by the matrix:
            [1/3   0]
            [  0 1/2]
            Domain: Vector space of dimension 2 over Rational Field
            Codomain: Vector space of dimension 2 over Rational Field

        Check that a certain non-invertible morphism isn't invertible::

            sage: V = ZZ^2; phi = V.hom([3*V.0, 2*V.1])
            sage: phi^(-1)
            Traceback (most recent call last):
            ...
            ZeroDivisionError: matrix morphism not invertible
        """
        try:
            B = ~(self.matrix())
        except ZeroDivisionError:
            raise ZeroDivisionError("matrix morphism not invertible")
        try:
            return self.parent().reversed()(B)
        except TypeError:
            raise ZeroDivisionError("matrix morphism not invertible")

    def inverse(self):
        r"""
        Returns the inverse of this matrix morphism, if the inverse exists.

        Raises a ``ZeroDivisionError`` if the inverse does not exist.

        EXAMPLES:

        An invertible morphism created as a restriction of
        a non-invertible morphism, and which has an unequal
        domain and codomain.  ::

            sage: V = QQ^4
            sage: W = QQ^3
            sage: m = matrix(QQ, [[2, 0, 3], [-6, 1, 4], [1, 2, -4], [1, 0, 1]])
            sage: phi = V.hom(m, W)
            sage: rho = phi.restrict_domain(V.span([V.0, V.3]))
            sage: zeta = rho.restrict_codomain(W.span([W.0, W.2]))
            sage: x = vector(QQ, [2, 0, 0, -7])
            sage: y = zeta(x); y
            (-3, 0, -1)
            sage: inv = zeta.inverse(); inv
            Vector space morphism represented by the matrix:
            [-1  3]
            [ 1 -2]
            Domain: Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [1 0 0]
            [0 0 1]
            Codomain: Vector space of degree 4 and dimension 2 over Rational Field
            Basis matrix:
            [1 0 0 0]
            [0 0 0 1]
            sage: inv(y) == x
            True

        An example of an invertible morphism between modules,
        (rather than between vector spaces).  ::

            sage: M = ZZ^4
            sage: p = matrix(ZZ, [[ 0, -1,  1, -2],
            ....:                 [ 1, -3,  2, -3],
            ....:                 [ 0,  4, -3,  4],
            ....:                 [-2,  8, -4,  3]])
            sage: phi = M.hom(p, M)
            sage: x = vector(ZZ, [1, -3, 5, -2])
            sage: y = phi(x); y
            (1, 12, -12, 21)
            sage: rho = phi.inverse(); rho
            Free module morphism defined by the matrix
            [ -5   3  -1   1]
            [ -9   4  -3   2]
            [-20   8  -7   4]
            [ -6   2  -2   1]
            Domain: Ambient free module of rank 4 over the principal ideal domain ...
            Codomain: Ambient free module of rank 4 over the principal ideal domain ...
            sage: rho(y) == x
            True

        A non-invertible morphism, despite having an appropriate
        domain and codomain.  ::

            sage: V = QQ^2
            sage: m = matrix(QQ, [[1, 2], [20, 40]])
            sage: phi = V.hom(m, V)
            sage: phi.is_bijective()
            False
            sage: phi.inverse()
            Traceback (most recent call last):
            ...
            ZeroDivisionError: matrix morphism not invertible

        The matrix representation of this morphism is invertible
        over the rationals, but not over the integers, thus the
        morphism is not invertible as a map between modules.
        It is easy to notice from the definition that every
        vector of the image will have a second entry that
        is an even integer.  ::

            sage: V = ZZ^2
            sage: q = matrix(ZZ, [[1, 2], [3, 4]])
            sage: phi = V.hom(q, V)
            sage: phi.matrix().change_ring(QQ).inverse()
            [  -2    1]
            [ 3/2 -1/2]
            sage: phi.is_bijective()
            False
            sage: phi.image()
            Free module of degree 2 and rank 2 over Integer Ring
            Echelon basis matrix:
            [1 0]
            [0 2]
            sage: phi.lift(vector(ZZ, [1, 1]))
            Traceback (most recent call last):
            ...
            ValueError: element is not in the image
            sage: phi.inverse()
            Traceback (most recent call last):
            ...
            ZeroDivisionError: matrix morphism not invertible

        The unary invert operator (~, tilde, "wiggle") is synonymous
        with the ``inverse()`` method (and a lot easier to type).  ::

            sage: V = QQ^2
            sage: r = matrix(QQ, [[4, 3], [-2, 5]])
            sage: phi = V.hom(r, V)
            sage: rho = phi.inverse()
            sage: zeta = ~phi
            sage: rho.is_equal_function(zeta)
            True

        TESTS::

            sage: V = QQ^2
            sage: W = QQ^3
            sage: U = W.span([W.0, W.1])
            sage: m = matrix(QQ, [[2, 1], [3, 4]])
            sage: phi = V.hom(m, U)
            sage: inv = phi.inverse()
            sage: (inv*phi).is_identity()
            True
            sage: (phi*inv).is_identity()
            True
        """
        return ~self

    def __rmul__(self, left):
        """
        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom([V.0+V.1, 2*V.1])
            sage: 2*phi
            Free module morphism defined by the matrix
            [2 2]
            [0 4]...
            sage: phi*2
            Free module morphism defined by the matrix
            [2 2]
            [0 4]...
        """
        R = self.base_ring()
        return self.parent()(R(left) * self.matrix())

    def __mul__(self, right):
        """
        Composition of morphisms, denoted by \*.

        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom([V.0+V.1, 2*V.1])
            sage: phi*phi
            Free module morphism defined by the matrix
            [1 3]
            [0 4]
            Domain: Ambient free module of rank 2 over the principal ideal domain ...
            Codomain: Ambient free module of rank 2 over the principal ideal domain ...

            sage: V = QQ^3
            sage: E = V.endomorphism_ring()
            sage: phi = E(Matrix(QQ,3,range(9))) ; phi
            Vector space morphism represented by the matrix:
            [0 1 2]
            [3 4 5]
            [6 7 8]
            Domain: Vector space of dimension 3 over Rational Field
            Codomain: Vector space of dimension 3 over Rational Field
            sage: phi*phi
            Vector space morphism represented by the matrix:
            [ 15  18  21]
            [ 42  54  66]
            [ 69  90 111]
            Domain: Vector space of dimension 3 over Rational Field
            Codomain: Vector space of dimension 3 over Rational Field
            sage: phi.matrix()**2
            [ 15  18  21]
            [ 42  54  66]
            [ 69  90 111]

        ::

            sage: W = QQ**4
            sage: E_VW = V.Hom(W)
            sage: psi = E_VW(Matrix(QQ,3,4,range(12))) ; psi
            Vector space morphism represented by the matrix:
            [ 0  1  2  3]
            [ 4  5  6  7]
            [ 8  9 10 11]
            Domain: Vector space of dimension 3 over Rational Field
            Codomain: Vector space of dimension 4 over Rational Field
            sage: psi*phi
            Vector space morphism represented by the matrix:
            [ 20  23  26  29]
            [ 56  68  80  92]
            [ 92 113 134 155]
            Domain: Vector space of dimension 3 over Rational Field
            Codomain: Vector space of dimension 4 over Rational Field
            sage: phi*psi
            Traceback (most recent call last):
            ...
            TypeError: Incompatible composition of morphisms: domain of left morphism must be codomain of right.
            sage: phi.matrix()*psi.matrix()
            [ 20  23  26  29]
            [ 56  68  80  92]
            [ 92 113 134 155]

        Composite maps can be formed with matrix morphisms::

            sage: K.<a> = NumberField(x^2 + 23)
            sage: V, VtoK, KtoV = K.vector_space()
            sage: f = V.hom([V.0 - V.1, V.0 + V.1])*KtoV; f
            Composite map:
            From: Number Field in a with defining polynomial x^2 + 23
            To:   Vector space of dimension 2 over Rational Field
            Defn:   Isomorphism map:
                    From: Number Field in a with defining polynomial x^2 + 23
                    To:   Vector space of dimension 2 over Rational Field
                    then
                    Vector space morphism represented by the matrix:
                    [ 1 -1]
                    [ 1  1]
                    Domain: Vector space of dimension 2 over Rational Field
                    Codomain: Vector space of dimension 2 over Rational Field
            sage: f(a)
            (1, 1)
        """
        if not isinstance(right, MatrixMorphism):
            if isinstance(right, (sage.categories.morphism.Morphism, sage.categories.map.Map)):
                return sage.categories.map.Map.__mul__(self, right)
            R = self.base_ring()
            return self.parent()(self.matrix() * R(right))
        if self.domain() != right.codomain():
            raise TypeError("Incompatible composition of morphisms: domain of left morphism must be codomain of right.")
        M = right.matrix() * self.matrix()
        return right.domain().Hom(self.codomain())(M)

    def __add__(self, right):
        """
        Sum of morphisms, denoted by +.

        EXAMPLES::

            sage: phi = (ZZ**2).endomorphism_ring()(Matrix(ZZ,2,[2..5])) ; phi
            Free module morphism defined by the matrix
            [2 3]
            [4 5]
            Domain: Ambient free module of rank 2 over the principal ideal domain ...
            Codomain: Ambient free module of rank 2 over the principal ideal domain ...
            sage: phi + 3
            Free module morphism defined by the matrix
            [5 3]
            [4 8]
            Domain: Ambient free module of rank 2 over the principal ideal domain ...
            Codomain: Ambient free module of rank 2 over the principal ideal domain ...
            sage: phi + phi
            Free module morphism defined by the matrix
            [ 4  6]
            [ 8 10]
            Domain: Ambient free module of rank 2 over the principal ideal domain ...
            Codomain: Ambient free module of rank 2 over the principal ideal domain ...
            sage: psi = (ZZ**3).endomorphism_ring()(Matrix(ZZ,3,[22..30])) ; psi
            Free module morphism defined by the matrix
            [22 23 24]
            [25 26 27]
            [28 29 30]
            Domain: Ambient free module of rank 3 over the principal ideal domain ...
            Codomain: Ambient free module of rank 3 over the principal ideal domain ...
            sage: phi + psi
            Traceback (most recent call last):
            ...
            ValueError: a matrix from
            Full MatrixSpace of 3 by 3 dense matrices over Integer Ring
            cannot be converted to a matrix in
            Full MatrixSpace of 2 by 2 dense matrices over Integer Ring!
        """
        # TODO: move over to any coercion model!
        if not isinstance(right, MatrixMorphism):
            R = self.base_ring()
            return self.parent()(self.matrix() + R(right))
        if not right.parent() == self.parent():
            right = self.parent()(right)
        M = self.matrix() + right.matrix()
        return self.domain().Hom(right.codomain())(M)

    def __neg__(self):
        """
        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom([V.0+V.1, 2*V.1])
            sage: -phi
            Free module morphism defined by the matrix
            [-1 -1]
            [ 0 -2]...
        """
        return self.parent()(-self.matrix())

    def __sub__(self, other):
        """
        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom([V.0+V.1, 2*V.1])
            sage: phi - phi
            Free module morphism defined by the matrix
            [0 0]
            [0 0]...
        """
        # TODO: move over to any coercion model!
        if not isinstance(other, MatrixMorphism):
            R = self.base_ring()
            return self.parent()(self.matrix() - R(other))
        if not other.parent() == self.parent():
            other = self.parent()(other)
        return self.parent()(self.matrix() - other.matrix())

    def base_ring(self):
        """
        Return the base ring of self, that is, the ring over which self is
        given by a matrix.

        EXAMPLES::

            sage: sage.modules.matrix_morphism.MatrixMorphism((ZZ**2).endomorphism_ring(), Matrix(ZZ,2,[3..6])).base_ring()
            Integer Ring
        """
        return self.domain().base_ring()

    def characteristic_polynomial(self, var='x'):
        r"""
        Return the characteristic polynomial of this endomorphism.

        ``characteristic_polynomial`` and ``char_poly`` are the same method.

        INPUT:
            - var -- variable

        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom([V.0+V.1, 2*V.1])
            sage: phi.characteristic_polynomial()
            x^2 - 3*x + 2
            sage: phi.charpoly()
            x^2 - 3*x + 2
            sage: phi.matrix().charpoly()
            x^2 - 3*x + 2
            sage: phi.charpoly('T')
            T^2 - 3*T + 2
        """
        if not self.is_endomorphism():
            raise ArithmeticError("charpoly only defined for endomorphisms " +\
                    "(i.e., domain = range)")
        return self.matrix().charpoly(var)

    charpoly = characteristic_polynomial

    def decomposition(self, *args, **kwds):
        """
        Return decomposition of this endomorphism, i.e., sequence of
        subspaces obtained by finding invariant subspaces of self.

        See the documentation for self.matrix().decomposition for more
        details.  All inputs to this function are passed onto the
        matrix one.

        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom([V.0+V.1, 2*V.1])
            sage: phi.decomposition()
            [
            Free module of degree 2 and rank 1 over Integer Ring
            Echelon basis matrix:
            [0 1],
            Free module of degree 2 and rank 1 over Integer Ring
            Echelon basis matrix:
            [ 1 -1]
            ]
        """
        if not self.is_endomorphism():
            raise ArithmeticError("Matrix morphism must be an endomorphism.")
        D = self.domain()
        E = self.matrix().decomposition(*args,**kwds)
        if D.is_ambient():
            return Sequence([D.submodule(V, check=False) for V, _ in E],
                            cr=True, check=False)
        else:
            B = D.basis_matrix()
            R = D.base_ring()
            return Sequence([D.submodule((V.basis_matrix() * B).row_module(R),
                                         check=False) for V, _ in E],
                            cr=True, check=False)

    def trace(self):
        r"""
        Return the trace of this endomorphism.

        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom([V.0+V.1, 2*V.1])
            sage: phi.trace()
            3
        """
        return self._matrix.trace()

    def det(self):
        """
        Return the determinant of this endomorphism.

        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom([V.0+V.1, 2*V.1])
            sage: phi.det()
            2
        """
        if not self.is_endomorphism():
            raise ArithmeticError("Matrix morphism must be an endomorphism.")
        return self.matrix().determinant()

    def fcp(self, var='x'):
        """
        Return the factorization of the characteristic polynomial.

        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom([V.0+V.1, 2*V.1])
            sage: phi.fcp()
            (x - 2) * (x - 1)
            sage: phi.fcp('T')
            (T - 2) * (T - 1)
        """
        return self.charpoly(var).factor()

    def kernel(self):
        """
        Compute the kernel of this morphism.

        EXAMPLES::

            sage: V = VectorSpace(QQ,3)
            sage: id = V.Hom(V)(identity_matrix(QQ,3))
            sage: null = V.Hom(V)(0*identity_matrix(QQ,3))
            sage: id.kernel()
            Vector space of degree 3 and dimension 0 over Rational Field
            Basis matrix:
            []
            sage: phi = V.Hom(V)(matrix(QQ,3,range(9)))
            sage: phi.kernel()
            Vector space of degree 3 and dimension 1 over Rational Field
            Basis matrix:
            [ 1 -2  1]
            sage: hom(CC^2, CC^2, matrix(CC, [[1,0], [0,1]])).kernel()
            Vector space of degree 2 and dimension 0 over Complex Field with 53 bits of precision
            Basis matrix:
            []
        """
        V = self.matrix().kernel()
        D = self.domain()
        if not D.is_ambient():
            # Transform V to ambient space
            # This is a matrix multiply:  we take the linear combinations of the basis for
            # D given by the elements of the basis for V.
            B = V.basis_matrix() * D.basis_matrix()
            V = B.row_module(D.base_ring())
        return self.domain().submodule(V, check=False)

    def image(self):
        """
        Compute the image of this morphism.

        EXAMPLES::

            sage: V = VectorSpace(QQ,3)
            sage: phi = V.Hom(V)(matrix(QQ, 3, range(9)))
            sage: phi.image()
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [ 1  0 -1]
            [ 0  1  2]
            sage: hom(GF(7)^3, GF(7)^2, zero_matrix(GF(7), 3, 2)).image()
            Vector space of degree 2 and dimension 0 over Finite Field of size 7
            Basis matrix:
            []


        Compute the image of the identity map on a ZZ-submodule::

            sage: V = (ZZ^2).span([[1,2],[3,4]])
            sage: phi = V.Hom(V)(identity_matrix(ZZ,2))
            sage: phi(V.0) == V.0
            True
            sage: phi(V.1) == V.1
            True
            sage: phi.image()
            Free module of degree 2 and rank 2 over Integer Ring
            Echelon basis matrix:
            [1 0]
            [0 2]
            sage: phi.image() == V
            True
        """
        V = self.matrix().image()
        D = self.codomain()
        if not D.is_ambient():
            # Transform V to ambient space
            # This is a matrix multiply:  we take the linear combinations of the basis for
            # D given by the elements of the basis for V.
            B = V.basis_matrix() * D.basis_matrix()
            V = B.row_module(self.domain().base_ring())
        return self.codomain().submodule(V, check=False)

    def matrix(self):
        """
        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom(V.basis())
            sage: phi.matrix()
            [1 0]
            [0 1]
            sage: sage.modules.matrix_morphism.MatrixMorphism_abstract.matrix(phi)
            Traceback (most recent call last):
            ...
            NotImplementedError: this method must be overridden in the extension class
        """
        raise NotImplementedError("this method must be overridden in the extension class")

    def _matrix_(self):
        """
        EXAMPLES:

        Check that this works with the :func:`matrix` function
        (:trac:`16844`)::

            sage: H = Hom(ZZ^2, ZZ^3)
            sage: x = H.an_element()
            sage: matrix(x)
            [0 0 0]
            [0 0 0]

        TESTS:

        ``matrix(x)`` is immutable::

            sage: H = Hom(QQ^3, QQ^2)
            sage: phi = H(matrix(QQ, 3, 2, list(reversed(range(6))))); phi
            Vector space morphism represented by the matrix:
            [5 4]
            [3 2]
            [1 0]
            Domain: Vector space of dimension 3 over Rational Field
            Codomain: Vector space of dimension 2 over Rational Field
            sage: A = phi.matrix()
            sage: A[1, 1] = 19
            Traceback (most recent call last):
            ...
            ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).
        """
        return self.matrix()

    def rank(self):
        r"""
        Returns the rank of the matrix representing this morphism.

        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom(V.basis())
            sage: phi.rank()
            2
            sage: V = ZZ^2; phi = V.hom([V.0, V.0])
            sage: phi.rank()
            1
        """
        return self.matrix().rank()

    def nullity(self):
        r"""
        Returns the nullity of the matrix representing this morphism, which is the
        dimension of its kernel.

        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom(V.basis())
            sage: phi.nullity()
            0
            sage: V = ZZ^2; phi = V.hom([V.0, V.0])
            sage: phi.nullity()
            1
        """
        return self._matrix.left_nullity()

    def is_bijective(self):
        r"""
        Tell whether ``self`` is bijective.

        EXAMPLES:

        Two morphisms that are obviously not bijective, simply on
        considerations of the dimensions.  However, each fullfills
        half of the requirements to be a bijection.  ::

            sage: V1 = QQ^2
            sage: V2 = QQ^3
            sage: m = matrix(QQ, [[1, 2, 3], [4, 5, 6]])
            sage: phi = V1.hom(m, V2)
            sage: phi.is_injective()
            True
            sage: phi.is_bijective()
            False
            sage: rho = V2.hom(m.transpose(), V1)
            sage: rho.is_surjective()
            True
            sage: rho.is_bijective()
            False

        We construct a simple bijection between two one-dimensional
        vector spaces.  ::

            sage: V1 = QQ^3
            sage: V2 = QQ^2
            sage: phi = V1.hom(matrix(QQ, [[1, 2], [3, 4], [5, 6]]), V2)
            sage: x = vector(QQ, [1, -1, 4])
            sage: y = phi(x); y
            (18, 22)
            sage: rho = phi.restrict_domain(V1.span([x]))
            sage: zeta = rho.restrict_codomain(V2.span([y]))
            sage: zeta.is_bijective()
            True

        AUTHOR:

        - Rob Beezer (2011-06-28)
        """
        return self.is_injective() and self.is_surjective()

    def is_identity(self):
        r"""
        Determines if this morphism is an identity function or not.

        EXAMPLES:

        A homomorphism that cannot possibly be the identity
        due to an unequal domain and codomain.  ::

            sage: V = QQ^3
            sage: W = QQ^2
            sage: m = matrix(QQ, [[1, 2], [3, 4], [5, 6]])
            sage: phi = V.hom(m, W)
            sage: phi.is_identity()
            False

        A bijection, but not the identity. ::

            sage: V = QQ^3
            sage: n = matrix(QQ, [[3, 1, -8], [5, -4, 6], [1, 1, -5]])
            sage: phi = V.hom(n, V)
            sage: phi.is_bijective()
            True
            sage: phi.is_identity()
            False

        A restriction that is the identity.  ::

            sage: V = QQ^3
            sage: p = matrix(QQ, [[1, 0, 0], [5, 8, 3], [0, 0, 1]])
            sage: phi = V.hom(p, V)
            sage: rho = phi.restrict(V.span([V.0, V.2]))
            sage: rho.is_identity()
            True

        An identity linear transformation that is defined with a
        domain and codomain with wildly different bases, so that the
        matrix representation is not simply the identity matrix. ::

            sage: A = matrix(QQ, [[1, 1, 0], [2, 3, -4], [2, 4, -7]])
            sage: B = matrix(QQ, [[2, 7, -2], [-1, -3, 1], [-1, -6, 2]])
            sage: U = (QQ^3).subspace_with_basis(A.rows())
            sage: V = (QQ^3).subspace_with_basis(B.rows())
            sage: H = Hom(U, V)
            sage: id = lambda x: x
            sage: phi = H(id)
            sage: phi([203, -179, 34])
            (203, -179, 34)
            sage: phi.matrix()
            [  1   0   1]
            [ -9 -18  -2]
            [-17 -31  -5]
            sage: phi.is_identity()
            True

        TESTS::

            sage: V = QQ^10
            sage: H = Hom(V, V)
            sage: id = H.identity()
            sage: id.is_identity()
            True

        AUTHOR:

        - Rob Beezer (2011-06-28)
        """
        if self.domain() != self.codomain():
            return False
        # testing for the identity matrix will only work for
        #   endomorphisms which have the same basis for domain and codomain
        #   so we test equality on a basis, which is sufficient
        return all( self(u) == u for u in self.domain().basis() )

    def is_zero(self):
        r"""
        Determines if this morphism is a zero function or not.

        EXAMPLES:

        A zero morphism created from a function.  ::

            sage: V = ZZ^5
            sage: W = ZZ^3
            sage: z = lambda x: zero_vector(ZZ, 3)
            sage: phi = V.hom(z, W)
            sage: phi.is_zero()
            True

        An image list that just barely makes a non-zero morphism.  ::

            sage: V = ZZ^4
            sage: W = ZZ^6
            sage: z = zero_vector(ZZ, 6)
            sage: images = [z, z, W.5, z]
            sage: phi = V.hom(images, W)
            sage: phi.is_zero()
            False

        TESTS::

            sage: V = QQ^10
            sage: W = QQ^3
            sage: H = Hom(V, W)
            sage: rho = H.zero()
            sage: rho.is_zero()
            True

        AUTHOR:

        - Rob Beezer (2011-07-15)
        """
        # any nonzero entry in any matrix representation
        #   disqualifies the morphism as having totally zero outputs
        return self._matrix.is_zero()

    def is_equal_function(self, other):
        r"""
        Determines if two morphisms are equal functions.

        INPUT:

        - ``other`` - a morphism to compare with ``self``

        OUTPUT:

        Returns ``True`` precisely when the two morphisms have
        equal domains and codomains (as sets) and produce identical
        output when given the same input.  Otherwise returns ``False``.

        This is useful when ``self`` and ``other`` may have different
        representations.

        Sage's default comparison of matrix morphisms requires the
        domains to have the same bases and the codomains to have the
        same bases, and then compares the matrix representations.
        This notion of equality is more permissive (it will
        return ``True`` "more often"), but is more correct
        mathematically.

        EXAMPLES:

        Three morphisms defined by combinations of different
        bases for the domain and codomain and different functions.
        Two are equal, the third is different from both of the others.  ::

            sage: B = matrix(QQ, [[-3,  5, -4,  2],
            ....:                 [-1,  2, -1,  4],
            ....:                 [ 4, -6,  5, -1],
            ....:                 [-5,  7, -6,  1]])
            sage: U = (QQ^4).subspace_with_basis(B.rows())
            sage: C = matrix(QQ, [[-1, -6, -4],
            ....:                 [ 3, -5,  6],
            ....:                 [ 1,  2,  3]])
            sage: V = (QQ^3).subspace_with_basis(C.rows())
            sage: H = Hom(U, V)

            sage: D = matrix(QQ, [[-7, -2, -5,  2],
            ....:                 [-5,  1, -4, -8],
            ....:                 [ 1, -1,  1,  4],
            ....:                 [-4, -1, -3,   1]])
            sage: X = (QQ^4).subspace_with_basis(D.rows())
            sage: E = matrix(QQ, [[ 4, -1,  4],
            ....:                 [ 5, -4, -5],
            ....:                 [-1,  0, -2]])
            sage: Y = (QQ^3).subspace_with_basis(E.rows())
            sage: K = Hom(X, Y)

            sage: f = lambda x: vector(QQ, [x[0]+x[1], 2*x[1]-4*x[2], 5*x[3]])
            sage: g = lambda x: vector(QQ, [x[0]-x[2], 2*x[1]-4*x[2], 5*x[3]])

            sage: rho = H(f)
            sage: phi = K(f)
            sage: zeta = H(g)

            sage: rho.is_equal_function(phi)
            True
            sage: phi.is_equal_function(rho)
            True
            sage: zeta.is_equal_function(rho)
            False
            sage: phi.is_equal_function(zeta)
            False

        TESTS::

            sage: H = Hom(ZZ^2, ZZ^2)
            sage: phi = H(matrix(ZZ, 2, range(4)))
            sage: phi.is_equal_function('junk')
            Traceback (most recent call last):
            ...
            TypeError: can only compare to a matrix morphism, not junk

        AUTHOR:

        - Rob Beezer (2011-07-15)
        """
        if not is_MatrixMorphism(other):
            msg = 'can only compare to a matrix morphism, not {0}'
            raise TypeError(msg.format(other))
        if self.domain() != other.domain():
            return False
        if self.codomain() != other.codomain():
            return False
        # check agreement on any basis of the domain
        return all( self(u) == other(u) for u in self.domain().basis() )

    def restrict_domain(self, sub):
        """
        Restrict this matrix morphism to a subspace sub of the domain. The
        subspace sub should have a basis() method and elements of the basis
        should be coercible into domain.

        The resulting morphism has the same codomain as before, but a new
        domain.

        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom([3*V.0, 2*V.1])
            sage: phi.restrict_domain(V.span([V.0]))
            Free module morphism defined by the matrix
            [3 0]
            Domain: Free module of degree 2 and rank 1 over Integer Ring
            Echelon ...
            Codomain: Ambient free module of rank 2 over the principal ideal domain ...
            sage: phi.restrict_domain(V.span([V.1]))
            Free module morphism defined by the matrix
            [0 2]...
        """
        D = self.domain()
        if hasattr(D, 'coordinate_module'):
            # We only have to do this in case the module supports
            # alternative basis.  Some modules do, some modules don't.
            V = D.coordinate_module(sub)
        else:
            V = sub.free_module()
        A = self.matrix().restrict_domain(V)
        H = sub.Hom(self.codomain())
        return H(A)

    def restrict_codomain(self, sub):
        """
        Restrict this matrix morphism to a subspace sub of the codomain.

        The resulting morphism has the same domain as before, but a new
        codomain.

        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom([4*(V.0+V.1),0])
            sage: W = V.span([2*(V.0+V.1)])
            sage: phi
            Free module morphism defined by the matrix
            [4 4]
            [0 0]
            Domain: Ambient free module of rank 2 over the principal ideal domain ...
            Codomain: Ambient free module of rank 2 over the principal ideal domain ...
            sage: psi = phi.restrict_codomain(W); psi
            Free module morphism defined by the matrix
            [2]
            [0]
            Domain: Ambient free module of rank 2 over the principal ideal domain ...
            Codomain: Free module of degree 2 and rank 1 over Integer Ring
            Echelon ...

        An example in which the codomain equals the full ambient space, but
        with a different basis::

            sage: V = QQ^2
            sage: W = V.span_of_basis([[1,2],[3,4]])
            sage: phi = V.hom(matrix(QQ,2,[1,0,2,0]),W)
            sage: phi.matrix()
            [1 0]
            [2 0]
            sage: phi(V.0)
            (1, 2)
            sage: phi(V.1)
            (2, 4)
            sage: X = V.span([[1,2]]); X
            Vector space of degree 2 and dimension 1 over Rational Field
            Basis matrix:
            [1 2]
            sage: phi(V.0) in X
            True
            sage: phi(V.1) in X
            True
            sage: psi = phi.restrict_codomain(X); psi
            Vector space morphism represented by the matrix:
            [1]
            [2]
            Domain: Vector space of dimension 2 over Rational Field
            Codomain: Vector space of degree 2 and dimension 1 over Rational Field
            Basis matrix:
            [1 2]
            sage: psi(V.0)
            (1, 2)
            sage: psi(V.1)
            (2, 4)
            sage: psi(V.0).parent() is X
            True
        """
        H = self.domain().Hom(sub)
        C = self.codomain()
        if hasattr(C, 'coordinate_module'):
            # We only have to do this in case the module supports
            # alternative basis.  Some modules do, some modules don't.
            V = C.coordinate_module(sub)
        else:
            V = sub.free_module()
        return H(self.matrix().restrict_codomain(V))


    def restrict(self, sub):
        """
        Restrict this matrix morphism to a subspace sub of the domain.

        The codomain and domain of the resulting matrix are both sub.

        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom([3*V.0, 2*V.1])
            sage: phi.restrict(V.span([V.0]))
            Free module morphism defined by the matrix
            [3]
            Domain: Free module of degree 2 and rank 1 over Integer Ring
            Echelon ...
            Codomain: Free module of degree 2 and rank 1 over Integer Ring
            Echelon ...

            sage: V = (QQ^2).span_of_basis([[1,2],[3,4]])
            sage: phi = V.hom([V.0+V.1, 2*V.1])
            sage: phi(V.1) == 2*V.1
            True
            sage: W = span([V.1])
            sage: phi(W)
            Vector space of degree 2 and dimension 1 over Rational Field
            Basis matrix:
            [  1 4/3]
            sage: psi = phi.restrict(W); psi
            Vector space morphism represented by the matrix:
            [2]
            Domain: Vector space of degree 2 and dimension 1 over Rational Field
            Basis matrix:
            [  1 4/3]
            Codomain: Vector space of degree 2 and dimension 1 over Rational Field
            Basis matrix:
            [  1 4/3]
            sage: psi.domain() == W
            True
            sage: psi(W.0) == 2*W.0
            True
        """
        if not self.is_endomorphism():
            raise ArithmeticError("matrix morphism must be an endomorphism")
        D = self.domain()
        C = self.codomain()
        if D is not C and (D.basis() != C.basis()):
            # Tricky case when two bases for same space
            return self.restrict_domain(sub).restrict_codomain(sub)
        if hasattr(D, 'coordinate_module'):
            # We only have to do this in case the module supports
            # alternative basis.  Some modules do, some modules don't.
            V = D.coordinate_module(sub)
        else:
            V = sub.free_module()
        A = self.matrix().restrict(V)
        H = sage.categories.homset.End(sub, self.domain().category())
        return H(A)


class MatrixMorphism(MatrixMorphism_abstract):
    """
    A morphism defined by a matrix.

    INPUT:

    -  ``parent`` -- a homspace

    -  ``A`` -- matrix or a :class:`MatrixMorphism_abstract` instance

    -  ``copy_matrix`` -- (default: ``True``) make an immutable copy of
       the matrix ``A`` if it is mutable; if ``False``, then this makes
       ``A`` immutable
    """
    def __init__(self, parent, A, copy_matrix=True):
        """
        Initialize ``self``.

        EXAMPLES::

            sage: from sage.modules.matrix_morphism import MatrixMorphism
            sage: T = End(ZZ^3)
            sage: M = MatrixSpace(ZZ,3)
            sage: I = M.identity_matrix()
            sage: A = MatrixMorphism(T, I)
            sage: loads(A.dumps()) == A
            True
        """
        if parent is None:
            raise ValueError("no parent given when creating this matrix morphism")
        if isinstance(A, MatrixMorphism_abstract):
            A = A.matrix()
        R = A.base_ring()
        if A.nrows() != parent.domain().rank():
            raise ArithmeticError("number of rows of matrix (={}) must equal rank of domain (={})".format(A.nrows(), parent.domain().rank()))
        if A.ncols() != parent.codomain().rank():
                raise ArithmeticError("number of columns of matrix (={}) must equal rank of codomain (={})".format(A.ncols(), parent.codomain().rank()))
        if A.is_mutable():
            if copy_matrix:
                from copy import copy
                A = copy(A)
            A.set_immutable()
        self._matrix = A
        MatrixMorphism_abstract.__init__(self, parent)

    def matrix(self, side='left'):
        r"""
        Return a matrix that defines this morphism.

        INPUT:

        - ``side`` -- (default: ``'left'``) the side of the matrix
          where a vector is placed to effect the morphism (function)

        OUTPUT:

        A matrix which represents the morphism, relative to bases
        for the domain and codomain.  If the modules are provided
        with user bases, then the representation is relative to
        these bases.

        Internally, Sage represents a matrix morphism with the
        matrix multiplying a row vector placed to the left of the
        matrix.  If the option ``side='right'`` is used, then a
        matrix is returned that acts on a vector to the right of
        the matrix.  These two matrices are just transposes of
        each other and the difference is just a preference for
        the style of representation.

        EXAMPLES::

            sage: V = ZZ^2; W = ZZ^3
            sage: m = column_matrix([3*V.0 - 5*V.1, 4*V.0 + 2*V.1, V.0 + V.1])
            sage: phi = V.hom(m, W)
            sage: phi.matrix()
            [ 3  4  1]
            [-5  2  1]

            sage: phi.matrix(side='right')
            [ 3 -5]
            [ 4  2]
            [ 1  1]

        TESTS::

            sage: V = ZZ^2
            sage: phi = V.hom([3*V.0, 2*V.1])
            sage: phi.matrix(side='junk')
            Traceback (most recent call last):
            ...
            ValueError: side must be 'left' or 'right', not junk
        """
        if not side in ['left', 'right']:
            raise ValueError("side must be 'left' or 'right', not {0}".format(side))
        if side == 'left':
            return self._matrix
        else:
            return self._matrix.transpose()

    def is_injective(self):
        """
        Tell whether ``self`` is injective.

        EXAMPLES::

            sage: V1 = QQ^2
            sage: V2 = QQ^3
            sage: phi = V1.hom(Matrix([[1,2,3],[4,5,6]]),V2)
            sage: phi.is_injective()
            True
            sage: psi = V2.hom(Matrix([[1,2],[3,4],[5,6]]),V1)
            sage: psi.is_injective()
            False

        AUTHOR:

        -- Simon King (2010-05)
        """
        return self._matrix.kernel().dimension() == 0

    def is_surjective(self):
        r"""
        Tell whether ``self`` is surjective.

        EXAMPLES::

            sage: V1 = QQ^2
            sage: V2 = QQ^3
            sage: phi = V1.hom(Matrix([[1,2,3],[4,5,6]]), V2)
            sage: phi.is_surjective()
            False
            sage: psi = V2.hom(Matrix([[1,2],[3,4],[5,6]]), V1)
            sage: psi.is_surjective()
            True

        An example over a PID that is not `\ZZ`.  ::

            sage: R = PolynomialRing(QQ, 'x')
            sage: A = R^2
            sage: B = R^2
            sage: H = A.hom([B([x^2-1, 1]), B([x^2, 1])])
            sage: H.image()
            Free module of degree 2 and rank 2 over Univariate Polynomial Ring in x over Rational Field
            Echelon basis matrix:
            [ 1  0]
            [ 0 -1]
            sage: H.is_surjective()
            True

        This tests if :trac:`11552` is fixed. ::

            sage: V = ZZ^2
            sage: m = matrix(ZZ, [[1,2],[0,2]])
            sage: phi = V.hom(m, V)
            sage: phi.lift(vector(ZZ, [0, 1]))
            Traceback (most recent call last):
            ...
            ValueError: element is not in the image
            sage: phi.is_surjective()
            False

        AUTHORS:

        - Simon King (2010-05)
        - Rob Beezer (2011-06-28)
        """
        # Testing equality of free modules over PIDs is unreliable
        #   see Trac #11579 for explanation and status
        # We test if image equals codomain with two inclusions
        #   reverse inclusion of below is trivially true
        return self.codomain().is_submodule(self.image())

    def _repr_(self):
        r"""
        Return string representation of this matrix morphism.

        This will typically be overloaded in a derived class.

        EXAMPLES::

            sage: V = ZZ^2; phi = V.hom([3*V.0, 2*V.1])
            sage: sage.modules.matrix_morphism.MatrixMorphism._repr_(phi)
            'Morphism defined by the matrix\n[3 0]\n[0 2]'

            sage: phi._repr_()
            'Free module morphism defined by the matrix\n[3 0]\n[0 2]\nDomain: Ambient free module of rank 2 over the principal ideal domain Integer Ring\nCodomain: Ambient free module of rank 2 over the principal ideal domain Integer Ring'
        """
        return "Morphism defined by the matrix\n{0}".format(self.matrix())
