Morphisms of free modules¶
- AUTHORS:
William Stein: initial version
Miguel Marco (2010-06-19): added eigenvalues, eigenvectors and minpoly functions
-
class
sage.modules.free_module_morphism.
BaseIsomorphism1D
¶ Bases:
sage.categories.morphism.Morphism
An isomorphism between a ring and a free rank-1 module over the ring.
EXAMPLES:
sage: R.<x,y> = QQ[] sage: V, from_V, to_V = R.free_module(R) sage: from_V Isomorphism morphism: From: Ambient free module of rank 1 over the integral domain Multivariate Polynomial Ring in x, y over Rational Field To: Multivariate Polynomial Ring in x, y over Rational Field
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is_injective
()¶ EXAMPLES:
sage: R.<x,y> = QQ[] sage: V, from_V, to_V = R.free_module(R) sage: from_V.is_injective() True
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is_surjective
()¶ EXAMPLES:
sage: R.<x,y> = QQ[] sage: V, from_V, to_V = R.free_module(R) sage: from_V.is_surjective() True
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class
sage.modules.free_module_morphism.
BaseIsomorphism1D_from_FM
(parent, basis=None)¶ Bases:
sage.modules.free_module_morphism.BaseIsomorphism1D
An isomorphism to a ring from its 1-dimensional free module
INPUT:
parent
– the homsetbasis
– (default 1) an invertible element of the ring
EXAMPLES:
sage: R.<x> = QQ[[]] sage: V, from_V, to_V = R.free_module(R) sage: v = to_V(1+x); v (1 + x) sage: from_V(v) 1 + x sage: W, from_W, to_W = R.free_module(R, basis=(1-x)) sage: W is V True sage: w = to_W(1+x); w (1 - x^2) sage: from_W(w) 1 + x + O(x^20)
The basis vector has to be a unit so that the map is an isomorphism:
sage: W, from_W, to_W = R.free_module(R, basis=x) Traceback (most recent call last): ... ValueError: Basis element must be a unit
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class
sage.modules.free_module_morphism.
BaseIsomorphism1D_to_FM
(parent, basis=None)¶ Bases:
sage.modules.free_module_morphism.BaseIsomorphism1D
An isomorphism from a ring to its 1-dimensional free module
INPUT:
parent
– the homsetbasis
– (default 1) an invertible element of the ring
EXAMPLES:
sage: R = Zmod(8) sage: V, from_V, to_V = R.free_module(R) sage: v = to_V(2); v (2) sage: from_V(v) 2 sage: W, from_W, to_W = R.free_module(R, basis=3) sage: W is V True sage: w = to_W(2); w (6) sage: from_W(w) 2
The basis vector has to be a unit so that the map is an isomorphism:
sage: W, from_W, to_W = R.free_module(R, basis=4) Traceback (most recent call last): ... ValueError: Basis element must be a unit
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class
sage.modules.free_module_morphism.
FreeModuleMorphism
(parent, A)¶ Bases:
sage.modules.matrix_morphism.MatrixMorphism
INPUT:
parent
- a homspace in a (sub) category of free modulesA
- matrix
EXAMPLES:
sage: V = ZZ^3; W = span([[1,2,3],[-1,2,8]], ZZ) sage: phi = V.hom(matrix(ZZ,3,[1..9])) sage: type(phi) <class 'sage.modules.free_module_morphism.FreeModuleMorphism'>
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change_ring
(R)¶ Change the ring over which this morphism is defined. This changes the ring of the domain, codomain, and underlying matrix.
EXAMPLES:
sage: V0 = span([[0,0,1],[0,2,0]],ZZ); V1 = span([[1/2,0],[0,2]],ZZ); W = span([[1,0],[0,6]],ZZ) sage: h = V0.hom([-3*V1.0-3*V1.1, -3*V1.0-3*V1.1]) sage: h.base_ring() Integer Ring sage: h Free module morphism defined by the matrix [-3 -3] [-3 -3]... sage: h.change_ring(QQ).base_ring() Rational Field sage: f = h.change_ring(QQ); f Vector space morphism represented by the matrix: [-3 -3] [-3 -3] Domain: Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [0 1 0] [0 0 1] Codomain: Vector space of degree 2 and dimension 2 over Rational Field Basis matrix: [1 0] [0 1] sage: f = h.change_ring(GF(7)); f Vector space morphism represented by the matrix: [4 4] [4 4] Domain: Vector space of degree 3 and dimension 2 over Finite Field of size 7 Basis matrix: [0 1 0] [0 0 1] Codomain: Vector space of degree 2 and dimension 2 over Finite Field of size 7 Basis matrix: [1 0] [0 1]
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eigenspaces
(extend=True)¶ Compute a list of subspaces formed by eigenvectors of
self
.INPUT:
extend
– (default:True
) determines if field extensions should be considered
OUTPUT:
a list of pairs
(eigenvalue, eigenspace)
EXAMPLES:
sage: V = QQ^3 sage: h = V.hom([[1,0,0],[0,0,1],[0,-1,0]], V) sage: h.eigenspaces() [(1, Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 0 0]), (-1*I, Vector space of degree 3 and dimension 1 over Algebraic Field Basis matrix: [ 0 1 1*I]), (1*I, Vector space of degree 3 and dimension 1 over Algebraic Field Basis matrix: [ 0 1 -1*I])] sage: h.eigenspaces(extend=False) [(1, Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 0 0])] sage: h = V.hom([[2,1,0], [0,2,0], [0,0,-1]], V) sage: h.eigenspaces() [(-1, Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [0 0 1]), (2, Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [0 1 0])] sage: h = V.hom([[2,1,0], [0,2,0], [0,0,2]], V) sage: h.eigenspaces() [(2, Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [0 1 0] [0 0 1])]
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eigenvalues
(extend=True)¶ Returns a list with the eigenvalues of the endomorphism of vector spaces.
INPUT:
extend
– boolean (default: True) decides if base field extensions should be considered or not.
EXAMPLES:
We compute the eigenvalues of an endomorphism of \(\QQ^3\):
sage: V=QQ^3 sage: H=V.endomorphism_ring()([[1,-1,0],[-1,1,1],[0,3,1]]) sage: H.eigenvalues() [3, 1, -1]
Note the effect of the
extend
option:sage: V=QQ^2 sage: H=V.endomorphism_ring()([[0,-1],[1,0]]) sage: H.eigenvalues() [-1*I, 1*I] sage: H.eigenvalues(extend=False) []
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eigenvectors
(extend=True)¶ Computes the subspace of eigenvectors of a given eigenvalue.
INPUT:
extend
– boolean (default: True) decides if base field extensions should be considered or not.
OUTPUT:
A sequence of tuples. Each tuple contains an eigenvalue, a sequence with a basis of the corresponding subspace of eigenvectors, and the algebraic multiplicity of the eigenvalue.
EXAMPLES:
sage: V=(QQ^4).subspace([[0,2,1,4],[1,2,5,0],[1,1,1,1]]) sage: H=(V.Hom(V))(matrix(QQ, [[0,1,0],[-1,0,0],[0,0,3]])) sage: H.eigenvectors() [(3, [ (0, 0, 1, -6/7) ], 1), (-1*I, [ (1, 1*I, 0, -0.571428571428572? + 2.428571428571429?*I) ], 1), (1*I, [ (1, -1*I, 0, -0.571428571428572? - 2.428571428571429?*I) ], 1)] sage: H.eigenvectors(extend=False) [(3, [ (0, 0, 1, -6/7) ], 1)] sage: H1=(V.Hom(V))(matrix(QQ, [[2,1,0],[0,2,0],[0,0,3]])) sage: H1.eigenvectors() [(3, [ (0, 0, 1, -6/7) ], 1), (2, [ (0, 1, 0, 17/7) ], 2)] sage: H1.eigenvectors(extend=False) [(3, [ (0, 0, 1, -6/7) ], 1), (2, [ (0, 1, 0, 17/7) ], 2)]
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inverse_image
(V)¶ Given a submodule V of the codomain of self, return the inverse image of V under self, i.e., the biggest submodule of the domain of self that maps into V.
EXAMPLES:
We test computing inverse images over a field:
sage: V = QQ^3; W = span([[1,2,3],[-1,2,5/3]], QQ) sage: phi = V.hom(matrix(QQ,3,[1..9])) sage: phi.rank() 2 sage: I = phi.inverse_image(W); I Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 0] [ 0 1 -1/2] sage: phi(I.0) in W True sage: phi(I.1) in W True sage: W = phi.image() sage: phi.inverse_image(W) == V True
We test computing inverse images between two spaces embedded in different ambient spaces.:
sage: V0 = span([[0,0,1],[0,2,0]],ZZ); V1 = span([[1/2,0],[0,2]],ZZ); W = span([[1,0],[0,6]],ZZ) sage: h = V0.hom([-3*V1.0-3*V1.1, -3*V1.0-3*V1.1]) sage: h.inverse_image(W) Free module of degree 3 and rank 2 over Integer Ring Echelon basis matrix: [0 2 1] [0 0 2] sage: h(h.inverse_image(W)).is_submodule(W) True sage: h(h.inverse_image(W)).index_in(W) +Infinity sage: h(h.inverse_image(W)) Free module of degree 2 and rank 1 over Integer Ring Echelon basis matrix: [ 3 12]
We test computing inverse images over the integers:
sage: V = QQ^3; W = V.span_of_basis([[2,2,3],[-1,2,5/3]], ZZ) sage: phi = W.hom([W.0, W.0-W.1]) sage: Z = W.span([2*W.1]); Z Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [ 2 -4 -10/3] sage: Y = phi.inverse_image(Z); Y Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [ 6 0 8/3] sage: phi(Y) == Z True
We test that trac ticket #24590 is resolved:
sage: A = FreeQuadraticModule(ZZ,1,matrix([2])) sage: f = A.Hom(A).an_element() sage: f.inverse_image(A) Free module of degree 1 and rank 1 over Integer Ring Echelon basis matrix: [1]
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lift
(x)¶ Given an element of the image, return an element of the codomain that maps onto it.
Note that
lift
andpreimage_representative
are equivalent names for this method, with the latter suggesting that the return value is a coset representative of the domain modulo the kernel of the morphism.EXAMPLES:
sage: X = QQ**2 sage: V = X.span([[2, 0], [0, 8]], ZZ) sage: W = (QQ**1).span([[1/12]], ZZ) sage: f = V.hom([W([1/3]), W([1/2])], W) sage: l=f.lift([1/3]); l # random (8, -16) sage: f(l) (1/3) sage: f(f.lift([1/2])) (1/2) sage: f(f.lift([1/6])) (1/6) sage: f.lift([1/12]) Traceback (most recent call last): ... ValueError: element is not in the image sage: f.lift([1/24]) Traceback (most recent call last): ... TypeError: element [1/24] is not in free module
This works for vector spaces, too:
sage: V = VectorSpace(GF(3), 2) sage: W = VectorSpace(GF(3), 3) sage: f = V.hom([W.1, W.1 - W.0]) sage: f.lift(W.1) (1, 0) sage: f.lift(W.2) Traceback (most recent call last): ... ValueError: element is not in the image sage: w = W((17, -2, 0)) sage: f(f.lift(w)) == w True
This example illustrates the use of the
preimage_representative
as an equivalent name for this method.sage: V = ZZ^3 sage: W = ZZ^2 sage: w = vector(ZZ, [1,2]) sage: f = V.hom([w, w, w], W) sage: f.preimage_representative(vector(ZZ, [10, 20])) (0, 0, 10)
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minimal_polynomial
(var='x')¶ Computes the minimal polynomial.
minpoly()
andminimal_polynomial()
are the same method.INPUT:
var
- string (default: ‘x’) a variable name
OUTPUT:
polynomial in var - the minimal polynomial of the endomorphism.
EXAMPLES:
Compute the minimal polynomial, and check it.
sage: V=GF(7)^3 sage: H=V.Hom(V)([[0,1,2],[-1,0,3],[2,4,1]]) sage: H Vector space morphism represented by the matrix: [0 1 2] [6 0 3] [2 4 1] Domain: Vector space of dimension 3 over Finite Field of size 7 Codomain: Vector space of dimension 3 over Finite Field of size 7 sage: H.minpoly() x^3 + 6*x^2 + 6*x + 1 sage: H.minimal_polynomial() x^3 + 6*x^2 + 6*x + 1 sage: H^3 + (H^2)*6 + H*6 + 1 Vector space morphism represented by the matrix: [0 0 0] [0 0 0] [0 0 0] Domain: Vector space of dimension 3 over Finite Field of size 7 Codomain: Vector space of dimension 3 over Finite Field of size 7
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minpoly
(var='x')¶ Computes the minimal polynomial.
minpoly()
andminimal_polynomial()
are the same method.INPUT:
var
- string (default: ‘x’) a variable name
OUTPUT:
polynomial in var - the minimal polynomial of the endomorphism.
EXAMPLES:
Compute the minimal polynomial, and check it.
sage: V=GF(7)^3 sage: H=V.Hom(V)([[0,1,2],[-1,0,3],[2,4,1]]) sage: H Vector space morphism represented by the matrix: [0 1 2] [6 0 3] [2 4 1] Domain: Vector space of dimension 3 over Finite Field of size 7 Codomain: Vector space of dimension 3 over Finite Field of size 7 sage: H.minpoly() x^3 + 6*x^2 + 6*x + 1 sage: H.minimal_polynomial() x^3 + 6*x^2 + 6*x + 1 sage: H^3 + (H^2)*6 + H*6 + 1 Vector space morphism represented by the matrix: [0 0 0] [0 0 0] [0 0 0] Domain: Vector space of dimension 3 over Finite Field of size 7 Codomain: Vector space of dimension 3 over Finite Field of size 7
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preimage_representative
(x)¶ Given an element of the image, return an element of the codomain that maps onto it.
Note that
lift
andpreimage_representative
are equivalent names for this method, with the latter suggesting that the return value is a coset representative of the domain modulo the kernel of the morphism.EXAMPLES:
sage: X = QQ**2 sage: V = X.span([[2, 0], [0, 8]], ZZ) sage: W = (QQ**1).span([[1/12]], ZZ) sage: f = V.hom([W([1/3]), W([1/2])], W) sage: l=f.lift([1/3]); l # random (8, -16) sage: f(l) (1/3) sage: f(f.lift([1/2])) (1/2) sage: f(f.lift([1/6])) (1/6) sage: f.lift([1/12]) Traceback (most recent call last): ... ValueError: element is not in the image sage: f.lift([1/24]) Traceback (most recent call last): ... TypeError: element [1/24] is not in free module
This works for vector spaces, too:
sage: V = VectorSpace(GF(3), 2) sage: W = VectorSpace(GF(3), 3) sage: f = V.hom([W.1, W.1 - W.0]) sage: f.lift(W.1) (1, 0) sage: f.lift(W.2) Traceback (most recent call last): ... ValueError: element is not in the image sage: w = W((17, -2, 0)) sage: f(f.lift(w)) == w True
This example illustrates the use of the
preimage_representative
as an equivalent name for this method.sage: V = ZZ^3 sage: W = ZZ^2 sage: w = vector(ZZ, [1,2]) sage: f = V.hom([w, w, w], W) sage: f.preimage_representative(vector(ZZ, [10, 20])) (0, 0, 10)
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pushforward
(x)¶ Compute the image of a sub-module of the domain.
EXAMPLES:
sage: V = QQ^3; W = span([[1,2,3],[-1,2,5/3]], QQ) sage: phi = V.hom(matrix(QQ,3,[1..9])) sage: phi.rank() 2 sage: phi(V) #indirect doctest Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 2]
We compute the image of a submodule of a ZZ-module embedded in a rational vector space:
sage: V = QQ^3; W = V.span_of_basis([[2,2,3],[-1,2,5/3]], ZZ) sage: phi = W.hom([W.0, W.0-W.1]); phi Free module morphism defined by the matrix [ 1 0] [ 1 -1]... sage: phi(span([2*W.1],ZZ)) Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [ 6 0 8/3] sage: phi(2*W.1) (6, 0, 8/3)
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sage.modules.free_module_morphism.
is_FreeModuleMorphism
(x)¶ EXAMPLES:
sage: V = ZZ^2; f = V.hom([V.1,-2*V.0]) sage: sage.modules.free_module_morphism.is_FreeModuleMorphism(f) True sage: sage.modules.free_module_morphism.is_FreeModuleMorphism(0) False