X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=2404af03d7f249fec64caab63e9954aca91af7ab;hb=59177e7be6376f64fe24061d576a9fde77588d57;hp=835f76337f9449612bcd113de950afd011317107;hpb=5fcc3b1bdec67eec2d76cc39c79a25af00830e30;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index 835f763..2404af0 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -126,16 +126,125 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): 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: n = ZZ.random_element(2,10).abs() + sage: J = eja_rn(n) + sage: J.one().is_nilpotent() + False + sage: J = eja_ln(n) + sage: J.one().is_nilpotent() + False + + The additive identity is always nilpotent:: + + sage: set_random_seed() + sage: n = ZZ.random_element(2,10).abs() + sage: J = eja_rn(n) + sage: J.zero().is_nilpotent() + True + sage: J = eja_ln(n) + sage: J.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): @@ -181,69 +290,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return fda_elt.matrix().transpose() - 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(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 - - """ - # 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') - - def minimal_polynomial(self): """ EXAMPLES:: @@ -305,59 +351,79 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return elt.minimal_polynomial() - def is_nilpotent(self): + def span_of_powers(self): """ - Return whether or not some power of this element is zero. + 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()) ) - 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: + def subalgebra_generated_by(self): + """ + Return the associative subalgebra of the parent EJA generated + by this element. - The identity element is never nilpotent:: + TESTS:: sage: set_random_seed() - sage: n = ZZ.random_element(2,10).abs() + sage: n = ZZ.random_element(1,10).abs() sage: J = eja_rn(n) - sage: J.one().is_nilpotent() - False + sage: x = J.random_element() + sage: x.subalgebra_generated_by().is_associative() + True sage: J = eja_ln(n) - sage: J.one().is_nilpotent() - False + sage: x = J.random_element() + sage: x.subalgebra_generated_by().is_associative() + True - The additive identity is always nilpotent:: + Squaring in the subalgebra should be the same thing as + squaring in the superalgebra:: - sage: set_random_seed() - sage: n = ZZ.random_element(2,10).abs() - sage: J = eja_rn(n) - sage: J.zero().is_nilpotent() - True - sage: J = eja_ln(n) - sage: J.zero().is_nilpotent() + 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 """ - # 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): @@ -422,9 +488,24 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): 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):