class Element(FiniteDimensionalAlgebraElement):
"""
An element of a Euclidean Jordan algebra.
-
- Since EJAs are commutative, the "right multiplication" matrix is
- also the left multiplication matrix and must be symmetric::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(1,10).abs()
- sage: J = eja_rn(5)
- sage: J.random_element().matrix().is_symmetric()
- True
- sage: J = eja_ln(5)
- sage: J.random_element().matrix().is_symmetric()
- True
-
"""
def __pow__(self, n):
EXAMPLES:
sage: set_random_seed()
- sage: J = eja_ln(5)
- sage: x = J.random_element()
+ sage: x = random_eja().random_element()
sage: x.matrix()*x.vector() == (x**2).vector()
True
return A.element_class(A, (self.matrix()**(n-1))*self.vector())
- def span_of_powers(self):
+ def characteristic_polynomial(self):
"""
- Return the vector space spanned by successive powers of
- this element.
+ Return my characteristic polynomial (if I'm a regular
+ element).
+
+ Eventually this should be implemented in terms of the parent
+ algebra's characteristic polynomial that works for ALL
+ elements.
"""
- # The dimension of the subalgebra can't be greater than
- # the big algebra, so just put everything into a list
- # and let span() get rid of the excess.
- V = self.vector().parent()
- return V.span( (self**d).vector() for d in xrange(V.dimension()) )
+ if self.is_regular():
+ return self.minimal_polynomial()
+ else:
+ raise NotImplementedError('irregular element')
+
+
+ def det(self):
+ """
+ Return my determinant, the product of my eigenvalues.
+
+ EXAMPLES::
+
+ sage: J = eja_ln(2)
+ sage: e0,e1 = J.gens()
+ sage: x = e0 + e1
+ sage: x.det()
+ 0
+ sage: J = eja_ln(3)
+ sage: e0,e1,e2 = J.gens()
+ sage: x = e0 + e1 + e2
+ sage: x.det()
+ -1
+
+ """
+ cs = self.characteristic_polynomial().coefficients(sparse=False)
+ r = len(cs) - 1
+ if r >= 0:
+ return cs[0] * (-1)**r
+ else:
+ raise ValueError('charpoly had no coefficients')
+
+
+ def is_nilpotent(self):
+ """
+ Return whether or not some power of this element is zero.
+
+ The superclass method won't work unless we're in an
+ associative algebra, and we aren't. However, we generate
+ an assocoative subalgebra and we're nilpotent there if and
+ only if we're nilpotent here (probably).
+
+ TESTS:
+
+ The identity element is never nilpotent::
+
+ sage: set_random_seed()
+ sage: random_eja().one().is_nilpotent()
+ False
+
+ The additive identity is always nilpotent::
+
+ sage: set_random_seed()
+ sage: random_eja().zero().is_nilpotent()
+ True
+
+ """
+ # The element we're going to call "is_nilpotent()" on.
+ # Either myself, interpreted as an element of a finite-
+ # dimensional algebra, or an element of an associative
+ # subalgebra.
+ elt = None
+
+ if self.parent().is_associative():
+ elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+ else:
+ V = self.span_of_powers()
+ assoc_subalg = self.subalgebra_generated_by()
+ # Mis-design warning: the basis used for span_of_powers()
+ # and subalgebra_generated_by() must be the same, and in
+ # the same order!
+ elt = assoc_subalg(V.coordinates(self.vector()))
+
+ # Recursive call, but should work since elt lives in an
+ # associative algebra.
+ return elt.is_nilpotent()
+
+
+ def is_regular(self):
+ """
+ Return whether or not this is a regular element.
+
+ EXAMPLES:
+
+ The identity element always has degree one, but any element
+ linearly-independent from it is regular::
+
+ sage: J = eja_ln(5)
+ sage: J.one().is_regular()
+ False
+ sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
+ sage: for x in J.gens():
+ ....: (J.one() + x).is_regular()
+ False
+ True
+ True
+ True
+ True
+
+ """
+ return self.degree() == self.parent().rank()
def degree(self):
return self.span_of_powers().dimension()
- def subalgebra_generated_by(self):
+ def matrix(self):
"""
- Return the associative subalgebra of the parent EJA generated
- by this element.
-
- TESTS::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(1,10).abs()
- sage: J = eja_rn(n)
- sage: x = J.random_element()
- sage: x.subalgebra_generated_by().is_associative()
- True
- sage: J = eja_ln(n)
- sage: x = J.random_element()
- sage: x.subalgebra_generated_by().is_associative()
- True
-
- Squaring in the subalgebra should be the same thing as
- squaring in the superalgebra::
-
- sage: J = eja_ln(5)
- sage: x = J.random_element()
- sage: u = x.subalgebra_generated_by().random_element()
- sage: u.matrix()*u.vector() == (u**2).vector()
- True
+ Return the matrix that represents left- (or right-)
+ multiplication by this element in the parent algebra.
+ We have to override this because the superclass method
+ returns a matrix that acts on row vectors (that is, on
+ the right).
"""
- # First get the subspace spanned by the powers of myself...
- V = self.span_of_powers()
- F = self.base_ring()
-
- # Now figure out the entries of the right-multiplication
- # matrix for the successive basis elements b0, b1,... of
- # that subspace.
- mats = []
- for b_right in V.basis():
- eja_b_right = self.parent()(b_right)
- b_right_rows = []
- # The first row of the right-multiplication matrix by
- # b1 is what we get if we apply that matrix to b1. The
- # second row of the right multiplication matrix by b1
- # is what we get when we apply that matrix to b2...
- #
- # IMPORTANT: this assumes that all vectors are COLUMN
- # vectors, unlike our superclass (which uses row vectors).
- for b_left in V.basis():
- eja_b_left = self.parent()(b_left)
- # Multiply in the original EJA, but then get the
- # coordinates from the subalgebra in terms of its
- # basis.
- this_row = V.coordinates((eja_b_left*eja_b_right).vector())
- b_right_rows.append(this_row)
- b_right_matrix = matrix(F, b_right_rows)
- mats.append(b_right_matrix)
-
- # It's an algebra of polynomials in one element, and EJAs
- # are power-associative.
- #
- # TODO: choose generator names intelligently.
- return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f')
+ fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+ return fda_elt.matrix().transpose()
def minimal_polynomial(self):
EXAMPLES::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,10).abs()
- sage: J = eja_rn(n)
- sage: x = J.random_element()
+ sage: x = random_eja().random_element()
sage: x.degree() == x.minimal_polynomial().degree()
True
::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,10).abs()
- sage: J = eja_ln(n)
- sage: x = J.random_element()
+ sage: x = random_eja().random_element()
sage: x.degree() == x.minimal_polynomial().degree()
True
return elt.minimal_polynomial()
- def is_nilpotent(self):
+ def quadratic_representation(self):
"""
- Return whether or not some power of this element is zero.
-
- The superclass method won't work unless we're in an
- associative algebra, and we aren't. However, we generate
- an assocoative subalgebra and we're nilpotent there if and
- only if we're nilpotent here (probably).
+ Return the quadratic representation of this element.
- TESTS:
+ EXAMPLES:
- The identity element is never nilpotent::
+ The explicit form in the spin factor algebra is given by
+ Alizadeh's Example 11.12::
- sage: set_random_seed()
- sage: n = ZZ.random_element(2,10).abs()
- sage: J = eja_rn(n)
- sage: J.one().is_nilpotent()
- False
+ sage: n = ZZ.random_element(1,10).abs()
sage: J = eja_ln(n)
- sage: J.one().is_nilpotent()
- False
+ sage: x = J.random_element()
+ sage: x_vec = x.vector()
+ sage: x0 = x_vec[0]
+ sage: x_bar = x_vec[1:]
+ sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
+ sage: B = 2*x0*x_bar.row()
+ sage: C = 2*x0*x_bar.column()
+ sage: D = identity_matrix(QQ, n-1)
+ sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
+ sage: D = D + 2*x_bar.tensor_product(x_bar)
+ sage: Q = block_matrix(2,2,[A,B,C,D])
+ sage: Q == x.quadratic_representation()
+ True
- The additive identity is always nilpotent::
+ """
+ return 2*(self.matrix()**2) - (self**2).matrix()
+
+
+ def span_of_powers(self):
+ """
+ Return the vector space spanned by successive powers of
+ this element.
+ """
+ # The dimension of the subalgebra can't be greater than
+ # the big algebra, so just put everything into a list
+ # and let span() get rid of the excess.
+ V = self.vector().parent()
+ return V.span( (self**d).vector() for d in xrange(V.dimension()) )
+
+
+ def subalgebra_generated_by(self):
+ """
+ Return the associative subalgebra of the parent EJA generated
+ by this element.
+
+ TESTS::
sage: set_random_seed()
- sage: n = ZZ.random_element(2,10).abs()
- sage: J = eja_rn(n)
- sage: J.zero().is_nilpotent()
+ sage: x = random_eja().random_element()
+ sage: x.subalgebra_generated_by().is_associative()
True
- sage: J = eja_ln(n)
- sage: J.zero().is_nilpotent()
+
+ Squaring in the subalgebra should be the same thing as
+ squaring in the superalgebra::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: u = x.subalgebra_generated_by().random_element()
+ sage: u.matrix()*u.vector() == (u**2).vector()
True
"""
- # The element we're going to call "is_nilpotent()" on.
- # Either myself, interpreted as an element of a finite-
- # dimensional algebra, or an element of an associative
- # subalgebra.
- elt = None
+ # First get the subspace spanned by the powers of myself...
+ V = self.span_of_powers()
+ F = self.base_ring()
- if self.parent().is_associative():
- elt = FiniteDimensionalAlgebraElement(self.parent(), self)
- else:
- V = self.span_of_powers()
- assoc_subalg = self.subalgebra_generated_by()
- # Mis-design warning: the basis used for span_of_powers()
- # and subalgebra_generated_by() must be the same, and in
- # the same order!
- elt = assoc_subalg(V.coordinates(self.vector()))
+ # Now figure out the entries of the right-multiplication
+ # matrix for the successive basis elements b0, b1,... of
+ # that subspace.
+ mats = []
+ for b_right in V.basis():
+ eja_b_right = self.parent()(b_right)
+ b_right_rows = []
+ # The first row of the right-multiplication matrix by
+ # b1 is what we get if we apply that matrix to b1. The
+ # second row of the right multiplication matrix by b1
+ # is what we get when we apply that matrix to b2...
+ #
+ # IMPORTANT: this assumes that all vectors are COLUMN
+ # vectors, unlike our superclass (which uses row vectors).
+ for b_left in V.basis():
+ eja_b_left = self.parent()(b_left)
+ # Multiply in the original EJA, but then get the
+ # coordinates from the subalgebra in terms of its
+ # basis.
+ this_row = V.coordinates((eja_b_left*eja_b_right).vector())
+ b_right_rows.append(this_row)
+ b_right_matrix = matrix(F, b_right_rows)
+ mats.append(b_right_matrix)
- # Recursive call, but should work since elt lives in an
- # associative algebra.
- return elt.is_nilpotent()
+ # It's an algebra of polynomials in one element, and EJAs
+ # are power-associative.
+ #
+ # TODO: choose generator names intelligently.
+ return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f')
def subalgebra_idempotent(self):
"""
Find an idempotent in the associative subalgebra I generate
using Proposition 2.3.5 in Baes.
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = eja_rn(5)
+ sage: c = J.random_element().subalgebra_idempotent()
+ sage: c^2 == c
+ True
+ sage: J = eja_ln(5)
+ sage: c = J.random_element().subalgebra_idempotent()
+ sage: c^2 == c
+ True
+
"""
if self.is_nilpotent():
raise ValueError("this only works with non-nilpotent elements!")
return self.parent().linear_combination(zip(c_coordinates, basis))
+ def trace(self):
+ """
+ Return my trace, the sum of my eigenvalues.
- def characteristic_polynomial(self):
- return self.matrix().characteristic_polynomial()
+ EXAMPLES::
+
+ sage: J = eja_ln(3)
+ sage: e0,e1,e2 = J.gens()
+ sage: x = e0 + e1 + e2
+ sage: x.trace()
+ 2
+
+ """
+ cs = self.characteristic_polynomial().coefficients(sparse=False)
+ if len(cs) >= 2:
+ return -1*cs[-2]
+ else:
+ raise ValueError('charpoly had fewer than 2 coefficients')
def eja_rn(dimension, field=QQ):
# ambient dimension).
rank = min(dimension,2)
return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=rank)
+
+
+def eja_sn(dimension, field=QQ):
+ """
+ Return the simple Jordan algebra of ``dimension``-by-``dimension``
+ symmetric matrices over ``field``.
+
+ EXAMPLES::
+
+ sage: J = eja_sn(2)
+ sage: e0, e1, e2 = J.gens()
+ sage: e0*e0
+ e0
+ sage: e1*e1
+ e0 + e2
+ sage: e2*e2
+ e2
+
+ """
+ Qs = []
+
+ # In S^2, for example, we nominally have four coordinates even
+ # though the space is of dimension three only. The vector space V
+ # is supposed to hold the entire long vector, and the subspace W
+ # of V will be spanned by the vectors that arise from symmetric
+ # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
+ V = VectorSpace(field, dimension**2)
+
+ # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
+ # coordinates.
+ S = []
+
+ for i in xrange(dimension):
+ for j in xrange(i+1):
+ Eij = matrix(field, dimension, lambda k,l: k==i and l==j)
+ if i == j:
+ Sij = Eij
+ else:
+ Sij = Eij + Eij.transpose()
+ S.append(Sij)
+
+ def mat2vec(m):
+ return vector(field, m.list())
+
+ def vec2mat(v):
+ return matrix(field, dimension, v.list())
+
+ W = V.span( mat2vec(s) for s in S )
+
+ # Taking the span above reorders our basis (thanks, jerk!) so we
+ # need to put our "matrix basis" in the same order as the
+ # (reordered) vector basis.
+ S = [ vec2mat(b) for b in W.basis() ]
+
+ for s in S:
+ # Brute force the multiplication-by-s matrix by looping
+ # through all elements of the basis and doing the computation
+ # to find out what the corresponding row should be. BEWARE:
+ # these multiplication tables won't be symmetric! It therefore
+ # becomes REALLY IMPORTANT that the underlying algebra
+ # constructor uses ROW vectors and not COLUMN vectors. That's
+ # why we're computing rows here and not columns.
+ Q_rows = []
+ for t in S:
+ this_row = mat2vec((s*t + t*s)/2)
+ Q_rows.append(W.coordinates(this_row))
+ Q = matrix(field,Q_rows)
+ Qs.append(Q)
+
+ return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension)
+
+
+def random_eja():
+ """
+ Return a "random" finite-dimensional Euclidean Jordan Algebra.
+
+ ALGORITHM:
+
+ For now, we choose a random natural number ``n`` (greater than zero)
+ and then give you back one of the following:
+
+ * The cartesian product of the rational numbers ``n`` times; this is
+ ``QQ^n`` with the Hadamard product.
+
+ * The Jordan spin algebra on ``QQ^n``.
+
+ * The ``n``-by-``n`` rational symmetric matrices with the symmetric
+ product.
+
+ Later this might be extended to return Cartesian products of the
+ EJAs above.
+
+ TESTS::
+
+ sage: random_eja()
+ Euclidean Jordan algebra of degree...
+
+ """
+ n = ZZ.random_element(1,10).abs()
+ constructor = choice([eja_rn, eja_ln, eja_sn])
+ return constructor(dimension=n, field=QQ)
projects / sage.d.git / blobdiff