X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=0fd1c5f2032151de7d752494b4117825af3109a3;hb=928b7d49fda98ff105c92293b5797bb7a2b9873a;hp=fbfd127ee496c46a9b8d06e3a1ca53129b8ce151;hpb=9550136d8a78dbc1d1aa891e31a66eb993d4be3e;p=sage.d.git diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index fbfd127..0fd1c5f 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -4,7 +4,8 @@ from sage.modules.free_module import VectorSpace from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement from mjo.eja.eja_operator import FiniteDimensionalEJAOperator -from mjo.eja.eja_utils import _mat2vec, _scale +from mjo.eja.eja_utils import _scale + class FiniteDimensionalEJAElement(IndexedFreeModuleElement): """ @@ -42,14 +43,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The definition of `x^2` is the unambiguous `x*x`:: - sage: set_random_seed() sage: x = random_eja().random_element() sage: x*x == (x^2) True A few examples of power-associativity:: - sage: set_random_seed() sage: x = random_eja().random_element() sage: x*(x*x)*(x*x) == x^5 True @@ -59,7 +58,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): We also know that powers operator-commute (Koecher, Chapter III, Corollary 1):: - sage: set_random_seed() sage: x = random_eja().random_element() sage: m = ZZ.random_element(0,10) sage: n = ZZ.random_element(0,10) @@ -106,7 +104,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): We should always get back an element of the algebra:: - sage: set_random_seed() sage: p = PolynomialRing(AA, 't').random_element() sage: J = random_eja() sage: x = J.random_element() @@ -131,7 +128,8 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): SETUP:: - sage: from mjo.eja.eja_algebra import HadamardEJA + sage: from mjo.eja.eja_algebra import (random_eja, + ....: HadamardEJA) EXAMPLES: @@ -155,7 +153,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The characteristic polynomial of an element should evaluate to zero on that element:: - sage: set_random_seed() sage: x = random_eja().random_element() sage: p = x.characteristic_polynomial() sage: x.apply_univariate_polynomial(p).is_zero() @@ -237,7 +234,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): Ensure that we can always compute an inner product, and that it gives us back a real number:: - sage: set_random_seed() sage: J = random_eja() sage: x,y = J.random_elements(2) sage: x.inner_product(y) in RLF @@ -265,7 +261,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The definition of a Jordan algebra says that any element operator-commutes with its square:: - sage: set_random_seed() sage: x = random_eja().random_element() sage: x.operator_commutes_with(x^2) True @@ -274,7 +269,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): Test Lemma 1 from Chapter III of Koecher:: - sage: set_random_seed() sage: u,v = random_eja().random_elements(2) sage: lhs = u.operator_commutes_with(u*v) sage: rhs = v.operator_commutes_with(u^2) @@ -284,7 +278,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): Test the first polarization identity from my notes, Koecher Chapter III, or from Baes (2.3):: - sage: set_random_seed() sage: x,y = random_eja().random_elements(2) sage: Lx = x.operator() sage: Ly = y.operator() @@ -296,7 +289,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): Test the second polarization identity from my notes or from Baes (2.4):: - sage: set_random_seed() sage: x,y,z = random_eja().random_elements(3) sage: Lx = x.operator() sage: Ly = y.operator() @@ -310,7 +302,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): Test the third polarization identity from my notes or from Baes (2.5):: - sage: set_random_seed() sage: u,y,z = random_eja().random_elements(3) sage: Lu = u.operator() sage: Ly = y.operator() @@ -375,7 +366,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): An element is invertible if and only if its determinant is non-zero:: - sage: set_random_seed() sage: x = random_eja().random_element() sage: x.is_invertible() == (x.det() != 0) True @@ -383,7 +373,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): Ensure that the determinant is multiplicative on an associative subalgebra as in Faraut and Korányi's Proposition II.2.2:: - sage: set_random_seed() sage: J = random_eja().random_element().subalgebra_generated_by() sage: x,y = J.random_elements(2) sage: (x*y).det() == x.det()*y.det() @@ -391,7 +380,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The determinant in real matrix algebras is the usual determinant:: - sage: set_random_seed() sage: X = matrix.random(QQ,3) sage: X = X + X.T sage: J1 = RealSymmetricEJA(3) @@ -448,7 +436,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The inverse in the spin factor algebra is given in Alizadeh's Example 11.11:: - sage: set_random_seed() sage: J = JordanSpinEJA.random_instance() sage: x = J.random_element() sage: while not x.is_invertible(): @@ -473,14 +460,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The identity element is its own inverse:: - sage: set_random_seed() sage: J = random_eja() sage: J.one().inverse() == J.one() True If an element has an inverse, it acts like one:: - sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: (not x.is_invertible()) or (x.inverse()*x == J.one()) @@ -488,7 +473,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The inverse of the inverse is what we started with:: - sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: (not x.is_invertible()) or (x.inverse().inverse() == x) @@ -498,7 +482,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): of an element is the inverse of its left-multiplication operator applied to the algebra's identity, when that inverse exists:: - sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: (not x.operator().is_invertible()) or ( @@ -508,7 +491,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): Check that the fast (cached) and slow algorithms give the same answer:: - sage: set_random_seed() # long time sage: J = random_eja(field=QQ, orthonormalize=False) # long time sage: x = J.random_element() # long time sage: while not x.is_invertible(): # long time @@ -560,14 +542,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The identity element is always invertible:: - sage: set_random_seed() sage: J = random_eja() sage: J.one().is_invertible() True The zero element is never invertible in a non-trivial algebra:: - sage: set_random_seed() sage: J = random_eja() sage: (not J.is_trivial()) and J.zero().is_invertible() False @@ -575,7 +555,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): Test that the fast (cached) and slow algorithms give the same answer:: - sage: set_random_seed() # long time sage: J = random_eja(field=QQ, orthonormalize=False) # long time sage: x = J.random_element() # long time sage: slow = x.is_invertible() # long time @@ -658,14 +637,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The identity element is minimal only in an EJA of rank one:: - sage: set_random_seed() sage: J = random_eja() sage: J.rank() == 1 or not J.one().is_primitive_idempotent() True A non-idempotent cannot be a minimal idempotent:: - sage: set_random_seed() sage: J = JordanSpinEJA(4) sage: x = J.random_element() sage: (not x.is_idempotent()) and x.is_primitive_idempotent() @@ -675,7 +652,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): idempotent if and only if it's idempotent with trace equal to unity:: - sage: set_random_seed() sage: J = JordanSpinEJA(4) sage: x = J.random_element() sage: expected = (x.is_idempotent() and x.trace() == 1) @@ -685,7 +661,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): Primitive idempotents must be non-zero:: - sage: set_random_seed() sage: J = random_eja() sage: J.zero().is_idempotent() True @@ -742,14 +717,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The identity element is never nilpotent, except in a trivial EJA:: - sage: set_random_seed() sage: J = random_eja() sage: J.one().is_nilpotent() and not J.is_trivial() False The additive identity is always nilpotent:: - sage: set_random_seed() sage: random_eja().zero().is_nilpotent() True @@ -792,7 +765,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The zero element should never be regular, unless the parent algebra has dimension less than or equal to one:: - sage: set_random_seed() sage: J = random_eja() sage: J.dimension() <= 1 or not J.zero().is_regular() True @@ -800,7 +772,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The unit element isn't regular unless the algebra happens to consist of only its scalar multiples:: - sage: set_random_seed() sage: J = random_eja() sage: J.dimension() <= 1 or not J.one().is_regular() True @@ -835,7 +806,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): In the spin factor algebra (of rank two), all elements that aren't multiples of the identity are regular:: - sage: set_random_seed() sage: J = JordanSpinEJA.random_instance() sage: n = J.dimension() sage: x = J.random_element() @@ -847,7 +817,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The zero and unit elements are both of degree one in nontrivial algebras:: - sage: set_random_seed() sage: J = random_eja() sage: d = J.zero().degree() sage: (J.is_trivial() and d == 0) or d == 1 @@ -858,7 +827,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): Our implementation agrees with the definition:: - sage: set_random_seed() sage: x = random_eja().random_element() sage: x.degree() == x.minimal_polynomial().degree() True @@ -967,7 +935,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): always the same, except in trivial algebras where the minimal polynomial of the unit/zero element is ``1``:: - sage: set_random_seed() sage: J = random_eja() sage: mu = J.one().minimal_polynomial() sage: t = mu.parent().gen() @@ -981,7 +948,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The degree of an element is (by one definition) the degree of its minimal polynomial:: - sage: set_random_seed() sage: x = random_eja().random_element() sage: x.degree() == x.minimal_polynomial().degree() True @@ -992,7 +958,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): identity. We require the dimension of the algebra to be at least two here so that said elements actually exist:: - sage: set_random_seed() sage: d_max = JordanSpinEJA._max_random_instance_dimension() sage: n = ZZ.random_element(2, max(2,d_max)) sage: J = JordanSpinEJA(n) @@ -1009,7 +974,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The minimal polynomial should always kill its element:: - sage: set_random_seed() sage: x = random_eja().random_element() sage: p = x.minimal_polynomial() sage: x.apply_univariate_polynomial(p) @@ -1018,9 +982,9 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The minimal polynomial is invariant under a change of basis, and in particular, a re-scaling of the basis:: - sage: set_random_seed() sage: d_max = RealSymmetricEJA._max_random_instance_dimension() - sage: n = ZZ.random_element(1, d_max) + sage: d = ZZ.random_element(1, d_max) + sage: n = RealSymmetricEJA._max_random_instance_size(d) sage: J1 = RealSymmetricEJA(n) sage: J2 = RealSymmetricEJA(n,orthonormalize=False) sage: X = random_matrix(AA,n) @@ -1119,18 +1083,10 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): B = self.parent().matrix_basis() W = self.parent().matrix_space() - if hasattr(W, 'cartesian_factors'): - # Aaaaand linear combinations don't work in Cartesian - # product spaces, even though they provide a method with - # that name. This is hidden behind an "if" because the - # _scale() function is slow. - pairs = zip(B, self.to_vector()) - return W.sum( _scale(b, alpha) for (b,alpha) in pairs ) - else: - # This is just a manual "from_vector()", but of course - # matrix spaces aren't vector spaces in sage, so they - # don't have a from_vector() method. - return W.linear_combination( zip(B, self.to_vector()) ) + # This is just a manual "from_vector()", but of course + # matrix spaces aren't vector spaces in sage, so they + # don't have a from_vector() method. + return W.linear_combination( zip(B, self.to_vector()) ) @@ -1172,7 +1128,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): TESTS:: - sage: set_random_seed() sage: J = random_eja() sage: x,y = J.random_elements(2) sage: x.operator()(y) == x*y @@ -1201,7 +1156,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The explicit form in the spin factor algebra is given by Alizadeh's Example 11.12:: - sage: set_random_seed() sage: x = JordanSpinEJA.random_instance().random_element() sage: x_vec = x.to_vector() sage: Q = matrix.identity(x.base_ring(), 0) @@ -1221,7 +1175,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): Test all of the properties from Theorem 11.2 in Alizadeh:: - sage: set_random_seed() sage: J = random_eja() sage: x,y = J.random_elements(2) sage: Lx = x.operator() @@ -1373,7 +1326,7 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): sage: (J0, J5, J1) = J.peirce_decomposition(c1) sage: (f0, f1, f2) = J1.gens() sage: f0.spectral_decomposition() - [(0, c2), (1, c0)] + [(0, 1.000000000000000?*c2), (1, 1.000000000000000?*c0)] """ A = self.subalgebra_generated_by(orthonormalize=True) @@ -1415,7 +1368,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): This subalgebra, being composed of only powers, is associative:: - sage: set_random_seed() sage: x0 = random_eja().random_element() sage: A = x0.subalgebra_generated_by() sage: x,y,z = A.random_elements(3) @@ -1425,7 +1377,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): Squaring in the subalgebra should work the same as in the superalgebra:: - sage: set_random_seed() sage: x = random_eja().random_element() sage: A = x.subalgebra_generated_by() sage: A(x^2) == A(x)*A(x) @@ -1436,7 +1387,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): element... unless the original algebra was trivial, in which case the subalgebra is trivial too:: - sage: set_random_seed() sage: A = random_eja().zero().subalgebra_generated_by() sage: (A.is_trivial() and A.dimension() == 0) or A.dimension() == 1 True @@ -1467,7 +1417,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): where there are non-nilpotent elements, or that we get the dumb solution in the trivial algebra:: - sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: while x.is_nilpotent() and not J.is_trivial(): @@ -1551,14 +1500,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The trace of an element is a real number:: - sage: set_random_seed() sage: J = random_eja() sage: J.random_element().trace() in RLF True The trace is linear:: - sage: set_random_seed() sage: J = random_eja() sage: x,y = J.random_elements(2) sage: alpha = J.base_ring().random_element() @@ -1594,7 +1541,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The trace inner product is commutative, bilinear, and associative:: - sage: set_random_seed() sage: J = random_eja() sage: x,y,z = J.random_elements(3) sage: # commutative @@ -1648,3 +1594,29 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): """ return self.trace_inner_product(self).sqrt() + + +class CartesianProductEJAElement(FiniteDimensionalEJAElement): + def det(self): + r""" + Compute the determinant of this product-element using the + determianants of its factors. + + This result Follows from the spectral decomposition of (say) + the pair `(x,y)` in terms of the Jordan frame `\left\{ (c_1, + 0),(c_2, 0),...,(0,d_1),(0,d_2),... \right\}. + """ + from sage.misc.misc_c import prod + return prod( f.det() for f in self.cartesian_factors() ) + + def to_matrix(self): + # An override is necessary to call our custom _scale(). + B = self.parent().matrix_basis() + W = self.parent().matrix_space() + + # Aaaaand linear combinations don't work in Cartesian + # product spaces, even though they provide a method with + # that name. This is hidden behind an "if" because the + # _scale() function is slow. + pairs = zip(B, self.to_vector()) + return W.sum( _scale(b, alpha) for (b,alpha) in pairs )