These functions are useful outside of Euclidean Jordan algebras.
It's looking in particular like the _all2list() gimmick may be
a useful general construct for algebras with a user basis.
from sage.combinat.free_module import CombinatorialFreeModule
from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
-from mjo.eja.eja_utils import _all2list
+from mjo.misc import _all2list
class UnitalClan(CombinatorialFreeModule):
Element = IndexedFreeModuleElement
from mjo.eja.eja_element import (CartesianProductEJAElement,
EJAElement)
from mjo.eja.eja_operator import EJAOperator
-from mjo.eja.eja_utils import _all2list
+from mjo.misc import _all2list
def EuclideanJordanAlgebras(field):
r"""
# at it, because we'd have to do it later anyway.
deortho_vector_basis = tuple( V(_all2list(b)) for b in basis )
- from mjo.eja.eja_utils import gram_schmidt
+ from mjo.misc import gram_schmidt
basis = tuple(gram_schmidt(basis, inner_product))
# Save the (possibly orthonormalized) matrix basis for
from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
from mjo.eja.eja_operator import EJAOperator
-from mjo.eja.eja_utils import _scale
+from mjo.misc import _scale
class EJAElement(IndexedFreeModuleElement):
+++ /dev/null
-def _scale(x, alpha):
- r"""
- Scale the vector, matrix, or cartesian-product-of-those-things
- ``x`` by ``alpha``.
-
- This works around the inability to scale certain elements of
- Cartesian product spaces, as reported in
-
- https://trac.sagemath.org/ticket/31435
-
- ..WARNING:
-
- This will do the wrong thing if you feed it a tuple or list.
-
- SETUP::
-
- sage: from mjo.eja.eja_utils import _scale
-
- EXAMPLES::
-
- sage: v = vector(QQ, (1,2,3))
- sage: _scale(v,2)
- (2, 4, 6)
- sage: m = matrix(QQ, [[1,2],[3,4]])
- sage: M = cartesian_product([m.parent(), m.parent()])
- sage: _scale(M((m,m)), 2)
- ([2 4]
- [6 8], [2 4]
- [6 8])
-
- """
- if hasattr(x, 'cartesian_factors'):
- P = x.parent()
- return P(tuple( _scale(x_i, alpha)
- for x_i in x.cartesian_factors() ))
- else:
- return x*alpha
-
-
-def _all2list(x):
- r"""
- Flatten a vector, matrix, or cartesian product of those things
- into a long list.
-
- If the entries of the matrix themselves belong to a real vector
- space (such as the complex numbers which can be thought of as
- pairs of real numbers), they will also be expanded in vector form
- and flattened into the list.
-
- SETUP::
-
- sage: from mjo.eja.eja_utils import _all2list
- sage: from mjo.hurwitz import (QuaternionMatrixAlgebra,
- ....: Octonions,
- ....: OctonionMatrixAlgebra)
-
- EXAMPLES::
-
- sage: _all2list([[1]])
- [1]
-
- ::
-
- sage: V1 = VectorSpace(QQ,2)
- sage: V2 = MatrixSpace(QQ,2)
- sage: x1 = V1([1,1])
- sage: x2 = V1([1,-1])
- sage: y1 = V2.one()
- sage: y2 = V2([0,1,1,0])
- sage: _all2list((x1,y1))
- [1, 1, 1, 0, 0, 1]
- sage: _all2list((x2,y2))
- [1, -1, 0, 1, 1, 0]
- sage: M = cartesian_product([V1,V2])
- sage: _all2list(M((x1,y1)))
- [1, 1, 1, 0, 0, 1]
- sage: _all2list(M((x2,y2)))
- [1, -1, 0, 1, 1, 0]
-
- ::
-
- sage: _all2list(Octonions().one())
- [1, 0, 0, 0, 0, 0, 0, 0]
- sage: _all2list(OctonionMatrixAlgebra(1).one())
- [1, 0, 0, 0, 0, 0, 0, 0]
-
- ::
-
- sage: _all2list(QuaternionAlgebra(QQ, -1, -1).one())
- [1, 0, 0, 0]
- sage: _all2list(QuaternionMatrixAlgebra(1).one())
- [1, 0, 0, 0]
-
- ::
-
- sage: V1 = VectorSpace(QQ,2)
- sage: V2 = OctonionMatrixAlgebra(1,field=QQ)
- sage: C = cartesian_product([V1,V2])
- sage: x1 = V1([3,4])
- sage: y1 = V2.one()
- sage: _all2list(C( (x1,y1) ))
- [3, 4, 1, 0, 0, 0, 0, 0, 0, 0]
-
- """
- if hasattr(x, 'to_vector'):
- # This works on matrices of e.g. octonions directly, without
- # first needing to convert them to a list of octonions and
- # then recursing down into the list. It also avoids the wonky
- # list(x) when x is an element of a CFM. I don't know what it
- # returns but it aint the coordinates. We don't recurse
- # because vectors can only contain ring elements as entries.
- return x.to_vector().list()
-
- from sage.structure.element import Matrix
- if isinstance(x, Matrix):
- # This sucks, but for performance reasons we don't want to
- # call _all2list recursively on the contents of a matrix
- # when we don't have to (they only contain ring elements
- # as entries)
- return x.list()
-
- try:
- xl = list(x)
- except TypeError: # x is not iterable
- return [x]
-
- if xl == [x]:
- # Avoid the retardation of list(QQ(1)) == [1].
- return [x]
-
- return sum( map(_all2list, xl) , [])
-
-
-def gram_schmidt(v, inner_product=None):
- """
- Perform Gram-Schmidt on the list ``v`` which are assumed to be
- vectors over the same base ring. Returns a list of orthonormalized
- vectors over the same base ring, which means that your base ring
- needs to contain the appropriate roots.
-
- SETUP::
-
- sage: from mjo.eja.eja_utils import gram_schmidt
-
- EXAMPLES:
-
- If you start with an orthonormal set, you get it back. We can use
- the rationals here because we don't need any square roots::
-
- sage: v1 = vector(QQ, (1,0,0))
- sage: v2 = vector(QQ, (0,1,0))
- sage: v3 = vector(QQ, (0,0,1))
- sage: v = [v1,v2,v3]
- sage: gram_schmidt(v) == v
- True
-
- The usual inner-product and norm are default::
-
- sage: v1 = vector(AA,(1,2,3))
- sage: v2 = vector(AA,(1,-1,6))
- sage: v3 = vector(AA,(2,1,-1))
- sage: v = [v1,v2,v3]
- sage: u = gram_schmidt(v)
- sage: all( u_i.inner_product(u_i).sqrt() == 1 for u_i in u )
- True
- sage: bool(u[0].inner_product(u[1]) == 0)
- True
- sage: bool(u[0].inner_product(u[2]) == 0)
- True
- sage: bool(u[1].inner_product(u[2]) == 0)
- True
-
-
- But if you supply a custom inner product, the result is
- orthonormal with respect to that (and not the usual inner
- product)::
-
- sage: v1 = vector(AA,(1,2,3))
- sage: v2 = vector(AA,(1,-1,6))
- sage: v3 = vector(AA,(2,1,-1))
- sage: v = [v1,v2,v3]
- sage: B = matrix(AA, [ [6, 4, 2],
- ....: [4, 5, 4],
- ....: [2, 4, 9] ])
- sage: ip = lambda x,y: (B*x).inner_product(y)
- sage: norm = lambda x: ip(x,x)
- sage: u = gram_schmidt(v,ip)
- sage: all( norm(u_i) == 1 for u_i in u )
- True
- sage: ip(u[0],u[1]).is_zero()
- True
- sage: ip(u[0],u[2]).is_zero()
- True
- sage: ip(u[1],u[2]).is_zero()
- True
-
- This Gram-Schmidt routine can be used on matrices as well, so long
- as an appropriate inner-product is provided::
-
- sage: E11 = matrix(AA, [ [1,0],
- ....: [0,0] ])
- sage: E12 = matrix(AA, [ [0,1],
- ....: [1,0] ])
- sage: E22 = matrix(AA, [ [0,0],
- ....: [0,1] ])
- sage: I = matrix.identity(AA,2)
- sage: trace_ip = lambda X,Y: (X*Y).trace()
- sage: gram_schmidt([E11,E12,I,E22], inner_product=trace_ip)
- [
- [1 0] [ 0 0.7071067811865475?] [0 0]
- [0 0], [0.7071067811865475? 0], [0 1]
- ]
-
- It even works on Cartesian product spaces whose factors are vector
- or matrix spaces::
-
- sage: V1 = VectorSpace(AA,2)
- sage: V2 = MatrixSpace(AA,2)
- sage: M = cartesian_product([V1,V2])
- sage: x1 = V1([1,1])
- sage: x2 = V1([1,-1])
- sage: y1 = V2.one()
- sage: y2 = V2([0,1,1,0])
- sage: z1 = M((x1,y1))
- sage: z2 = M((x2,y2))
- sage: def ip(a,b):
- ....: return a[0].inner_product(b[0]) + (a[1]*b[1]).trace()
- sage: U = gram_schmidt([z1,z2], inner_product=ip)
- sage: ip(U[0],U[1])
- 0
- sage: ip(U[0],U[0])
- 1
- sage: ip(U[1],U[1])
- 1
-
- TESTS:
-
- Ensure that zero vectors don't get in the way::
-
- sage: v1 = vector(AA,(1,2,3))
- sage: v2 = vector(AA,(1,-1,6))
- sage: v3 = vector(AA,(0,0,0))
- sage: v = [v1,v2,v3]
- sage: len(gram_schmidt(v)) == 2
- True
-
- """
- if len(v) == 0:
- # cool
- return v
-
- V = v[0].parent()
-
- if inner_product is None:
- inner_product = lambda x,y: x.inner_product(y)
-
- sc = lambda x,a: a*x
- if hasattr(V, 'cartesian_factors'):
- # Only use the slow implementation if necessary.
- sc = _scale
-
- def proj(x,y):
- # project y onto the span of {x}
- return sc(x, (inner_product(x,y)/inner_product(x,x)))
-
- def normalize(x):
- # Don't extend the given field with the necessary
- # square roots. This will probably throw weird
- # errors about the symbolic ring if you e.g. try
- # to use it on a set of rational vectors that isn't
- # already orthonormalized.
- return sc(x, ~inner_product(x,x).sqrt())
-
- v_out = [] # make a copy, don't clobber the input
-
- for (i, v_i) in enumerate(v):
- ortho_v_i = v_i - V.sum( proj(v_out[j],v_i) for j in range(i) )
- if not ortho_v_i.is_zero():
- v_out.append(normalize(ortho_v_i))
-
- return v_out
Stuff that doesn't fit anywhere else.
"""
+def _all2list(x):
+ r"""
+ Flatten a vector, matrix, or cartesian product of those things
+ into a long list.
+
+ If the entries of the matrix themselves belong to a real vector
+ space (such as the complex numbers which can be thought of as
+ pairs of real numbers), they will also be expanded in vector form
+ and flattened into the list.
+
+ SETUP::
+
+ sage: from mjo.misc import _all2list
+ sage: from mjo.hurwitz import (QuaternionMatrixAlgebra,
+ ....: Octonions,
+ ....: OctonionMatrixAlgebra)
+
+ EXAMPLES::
+
+ sage: _all2list([[1]])
+ [1]
+
+ ::
+
+ sage: V1 = VectorSpace(QQ,2)
+ sage: V2 = MatrixSpace(QQ,2)
+ sage: x1 = V1([1,1])
+ sage: x2 = V1([1,-1])
+ sage: y1 = V2.one()
+ sage: y2 = V2([0,1,1,0])
+ sage: _all2list((x1,y1))
+ [1, 1, 1, 0, 0, 1]
+ sage: _all2list((x2,y2))
+ [1, -1, 0, 1, 1, 0]
+ sage: M = cartesian_product([V1,V2])
+ sage: _all2list(M((x1,y1)))
+ [1, 1, 1, 0, 0, 1]
+ sage: _all2list(M((x2,y2)))
+ [1, -1, 0, 1, 1, 0]
+
+ ::
+
+ sage: _all2list(Octonions().one())
+ [1, 0, 0, 0, 0, 0, 0, 0]
+ sage: _all2list(OctonionMatrixAlgebra(1).one())
+ [1, 0, 0, 0, 0, 0, 0, 0]
+
+ ::
+
+ sage: _all2list(QuaternionAlgebra(QQ, -1, -1).one())
+ [1, 0, 0, 0]
+ sage: _all2list(QuaternionMatrixAlgebra(1).one())
+ [1, 0, 0, 0]
+
+ ::
+
+ sage: V1 = VectorSpace(QQ,2)
+ sage: V2 = OctonionMatrixAlgebra(1,field=QQ)
+ sage: C = cartesian_product([V1,V2])
+ sage: x1 = V1([3,4])
+ sage: y1 = V2.one()
+ sage: _all2list(C( (x1,y1) ))
+ [3, 4, 1, 0, 0, 0, 0, 0, 0, 0]
+
+ """
+ if hasattr(x, 'to_vector'):
+ # This works on matrices of e.g. octonions directly, without
+ # first needing to convert them to a list of octonions and
+ # then recursing down into the list. It also avoids the wonky
+ # list(x) when x is an element of a CFM. I don't know what it
+ # returns but it aint the coordinates. We don't recurse
+ # because vectors can only contain ring elements as entries.
+ return x.to_vector().list()
+
+ from sage.structure.element import Matrix
+ if isinstance(x, Matrix):
+ # This sucks, but for performance reasons we don't want to
+ # call _all2list recursively on the contents of a matrix
+ # when we don't have to (they only contain ring elements
+ # as entries)
+ return x.list()
+
+ try:
+ xl = list(x)
+ except TypeError: # x is not iterable
+ return [x]
+
+ if xl == [x]:
+ # Avoid the retardation of list(QQ(1)) == [1].
+ return [x]
+
+ return sum( map(_all2list, xl) , [])
+
+
+def gram_schmidt(v, inner_product=None):
+ """
+ Perform Gram-Schmidt on the list ``v`` which are assumed to be
+ vectors over the same base ring. Returns a list of orthonormalized
+ vectors over the same base ring, which means that your base ring
+ needs to contain the appropriate roots.
+
+ SETUP::
+
+ sage: from mjo.misc import gram_schmidt
+
+ EXAMPLES:
+
+ If you start with an orthonormal set, you get it back. We can use
+ the rationals here because we don't need any square roots::
+
+ sage: v1 = vector(QQ, (1,0,0))
+ sage: v2 = vector(QQ, (0,1,0))
+ sage: v3 = vector(QQ, (0,0,1))
+ sage: v = [v1,v2,v3]
+ sage: gram_schmidt(v) == v
+ True
+
+ The usual inner-product and norm are default::
+
+ sage: v1 = vector(AA,(1,2,3))
+ sage: v2 = vector(AA,(1,-1,6))
+ sage: v3 = vector(AA,(2,1,-1))
+ sage: v = [v1,v2,v3]
+ sage: u = gram_schmidt(v)
+ sage: all( u_i.inner_product(u_i).sqrt() == 1 for u_i in u )
+ True
+ sage: bool(u[0].inner_product(u[1]) == 0)
+ True
+ sage: bool(u[0].inner_product(u[2]) == 0)
+ True
+ sage: bool(u[1].inner_product(u[2]) == 0)
+ True
+
+
+ But if you supply a custom inner product, the result is
+ orthonormal with respect to that (and not the usual inner
+ product)::
+
+ sage: v1 = vector(AA,(1,2,3))
+ sage: v2 = vector(AA,(1,-1,6))
+ sage: v3 = vector(AA,(2,1,-1))
+ sage: v = [v1,v2,v3]
+ sage: B = matrix(AA, [ [6, 4, 2],
+ ....: [4, 5, 4],
+ ....: [2, 4, 9] ])
+ sage: ip = lambda x,y: (B*x).inner_product(y)
+ sage: norm = lambda x: ip(x,x)
+ sage: u = gram_schmidt(v,ip)
+ sage: all( norm(u_i) == 1 for u_i in u )
+ True
+ sage: ip(u[0],u[1]).is_zero()
+ True
+ sage: ip(u[0],u[2]).is_zero()
+ True
+ sage: ip(u[1],u[2]).is_zero()
+ True
+
+ This Gram-Schmidt routine can be used on matrices as well, so long
+ as an appropriate inner-product is provided::
+
+ sage: E11 = matrix(AA, [ [1,0],
+ ....: [0,0] ])
+ sage: E12 = matrix(AA, [ [0,1],
+ ....: [1,0] ])
+ sage: E22 = matrix(AA, [ [0,0],
+ ....: [0,1] ])
+ sage: I = matrix.identity(AA,2)
+ sage: trace_ip = lambda X,Y: (X*Y).trace()
+ sage: gram_schmidt([E11,E12,I,E22], inner_product=trace_ip)
+ [
+ [1 0] [ 0 0.7071067811865475?] [0 0]
+ [0 0], [0.7071067811865475? 0], [0 1]
+ ]
+
+ It even works on Cartesian product spaces whose factors are vector
+ or matrix spaces::
+
+ sage: V1 = VectorSpace(AA,2)
+ sage: V2 = MatrixSpace(AA,2)
+ sage: M = cartesian_product([V1,V2])
+ sage: x1 = V1([1,1])
+ sage: x2 = V1([1,-1])
+ sage: y1 = V2.one()
+ sage: y2 = V2([0,1,1,0])
+ sage: z1 = M((x1,y1))
+ sage: z2 = M((x2,y2))
+ sage: def ip(a,b):
+ ....: return a[0].inner_product(b[0]) + (a[1]*b[1]).trace()
+ sage: U = gram_schmidt([z1,z2], inner_product=ip)
+ sage: ip(U[0],U[1])
+ 0
+ sage: ip(U[0],U[0])
+ 1
+ sage: ip(U[1],U[1])
+ 1
+
+ TESTS:
+
+ Ensure that zero vectors don't get in the way::
+
+ sage: v1 = vector(AA,(1,2,3))
+ sage: v2 = vector(AA,(1,-1,6))
+ sage: v3 = vector(AA,(0,0,0))
+ sage: v = [v1,v2,v3]
+ sage: len(gram_schmidt(v)) == 2
+ True
+
+ """
+ if len(v) == 0:
+ # cool
+ return v
+
+ V = v[0].parent()
+
+ if inner_product is None:
+ inner_product = lambda x,y: x.inner_product(y)
+
+ sc = lambda x,a: a*x
+ if hasattr(V, 'cartesian_factors'):
+ # Only use the slow implementation if necessary.
+ sc = _scale
+
+ def proj(x,y):
+ # project y onto the span of {x}
+ return sc(x, (inner_product(x,y)/inner_product(x,x)))
+
+ def normalize(x):
+ # Don't extend the given field with the necessary
+ # square roots. This will probably throw weird
+ # errors about the symbolic ring if you e.g. try
+ # to use it on a set of rational vectors that isn't
+ # already orthonormalized.
+ return sc(x, ~inner_product(x,x).sqrt())
+
+ v_out = [] # make a copy, don't clobber the input
+
+ for (i, v_i) in enumerate(v):
+ ortho_v_i = v_i - V.sum( proj(v_out[j],v_i) for j in range(i) )
+ if not ortho_v_i.is_zero():
+ v_out.append(normalize(ortho_v_i))
+
+ return v_out
+
+
def legend_latex(obj):
"""
Return the LaTeX representation of ``obj``, but wrap it in dollar
"""
from sage.misc.latex import latex
return '$%s$' % latex(obj)
+
+
+def _scale(x, alpha):
+ r"""
+ Scale the vector, matrix, or cartesian-product-of-those-things
+ ``x`` by ``alpha``.
+
+ This works around the inability to scale certain elements of
+ Cartesian product spaces, as reported in
+
+ https://trac.sagemath.org/ticket/31435
+
+ ..WARNING:
+
+ This will do the wrong thing if you feed it a tuple or list.
+
+ SETUP::
+
+ sage: from mjo.misc import _scale
+
+ EXAMPLES::
+
+ sage: v = vector(QQ, (1,2,3))
+ sage: _scale(v,2)
+ (2, 4, 6)
+ sage: m = matrix(QQ, [[1,2],[3,4]])
+ sage: M = cartesian_product([m.parent(), m.parent()])
+ sage: _scale(M((m,m)), 2)
+ ([2 4]
+ [6 8], [2 4]
+ [6 8])
+
+ """
+ if hasattr(x, 'cartesian_factors'):
+ P = x.parent()
+ return P(tuple( _scale(x_i, alpha)
+ for x_i in x.cartesian_factors() ))
+ else:
+ return x*alpha
projects / sage.d.git / commitdiff