X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=f4d5995ce9302d2fdc518dac2b0ca20f0fadf6d1;hb=HEAD;hp=c98d9a2b64651562928a6489ef88e03fdd2b0dc9;hpb=bc02bf48592e22d034310cfffef8fb2a062c0a43;p=sage.d.git diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index c98d9a2..f4d5995 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -3,10 +3,11 @@ from sage.misc.cachefunc import cached_method 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_operator import EJAOperator +from mjo.eja.eja_utils import _scale -class FiniteDimensionalEJAElement(IndexedFreeModuleElement): + +class EJAElement(IndexedFreeModuleElement): """ An element of a Euclidean Jordan algebra. """ @@ -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: @@ -155,11 +153,10 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The characteristic polynomial of an element should evaluate to zero on that element:: - sage: set_random_seed() - sage: x = HadamardEJA(3).random_element() + sage: x = random_eja().random_element() sage: p = x.characteristic_polynomial() - sage: x.apply_univariate_polynomial(p) - 0 + sage: x.apply_univariate_polynomial(p).is_zero() + True The characteristic polynomials of the zero and unit elements should be what we think they are in a subalgebra, too:: @@ -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() @@ -296,32 +289,32 @@ 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() - sage: Lz = z.operator() - sage: Lzy = (z*y).operator() - sage: Lxy = (x*y).operator() - sage: Lxz = (x*z).operator() - sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly) + sage: x,y,z = random_eja().random_elements(3) # long time + sage: Lx = x.operator() # long time + sage: Ly = y.operator() # long time + sage: Lz = z.operator() # long time + sage: Lzy = (z*y).operator() # long time + sage: Lxy = (x*y).operator() # long time + sage: Lxz = (x*z).operator() # long time + sage: lhs = Lx*Lzy + Lz*Lxy + Ly*Lxz # long time + sage: rhs = Lzy*Lx + Lxy*Lz + Lxz*Ly # long time + sage: bool(lhs == rhs) # long time True 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() - sage: Lz = z.operator() - sage: Lzy = (z*y).operator() - sage: Luy = (u*y).operator() - sage: Luz = (u*z).operator() - sage: Luyz = (u*(y*z)).operator() - sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz - sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly - sage: bool(lhs == rhs) + sage: u,y,z = random_eja().random_elements(3) # long time + sage: Lu = u.operator() # long time + sage: Ly = y.operator() # long time + sage: Lz = z.operator() # long time + sage: Lzy = (z*y).operator() # long time + sage: Luy = (u*y).operator() # long time + sage: Luz = (u*z).operator() # long time + sage: Luyz = (u*(y*z)).operator() # long time + sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz # long time + sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly # long time + sage: bool(lhs == rhs) # long time True """ @@ -339,7 +332,8 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): SETUP:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + sage: from mjo.eja.eja_algebra import (AlbertEJA, + ....: JordanSpinEJA, ....: TrivialEJA, ....: RealSymmetricEJA, ....: ComplexHermitianEJA, @@ -375,7 +369,6 @@ 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 @@ -383,15 +376,14 @@ 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: x0 = random_eja().random_element() + sage: J = x0.subalgebra_generated_by(orthonormalize=False) sage: x,y = J.random_elements(2) sage: (x*y).det() == x.det()*y.det() True 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) @@ -404,6 +396,22 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): sage: actual2 == expected True + There's a formula for the determinant of the Albert algebra + (Yokota, Section 2.1):: + + sage: def albert_det(x): + ....: X = x.to_matrix() + ....: res = X[0,0]*X[1,1]*X[2,2] + ....: res += 2*(X[1,2]*X[2,0]*X[0,1]).real() + ....: res -= X[0,0]*X[1,2]*X[2,1] + ....: res -= X[1,1]*X[2,0]*X[0,2] + ....: res -= X[2,2]*X[0,1]*X[1,0] + ....: return res.leading_coefficient() + sage: J = AlbertEJA(field=QQ, orthonormalize=False) + sage: xs = J.random_elements(10) + sage: all( albert_det(x) == x.det() for x in xs ) + True + """ P = self.parent() r = P.rank() @@ -448,7 +456,6 @@ 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 +480,12 @@ 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 +493,6 @@ 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) @@ -498,17 +502,17 @@ 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 ( - ....: x.operator().inverse()(J.one()) == x.inverse() ) + sage: J = random_eja() # long time + sage: x = J.random_element() # long time + sage: (not x.operator().is_invertible()) or ( # long time + ....: x.operator().inverse()(J.one()) # long time + ....: == # long time + ....: x.inverse() ) # long time True 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 @@ -520,15 +524,18 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): True """ not_invertible_msg = "element is not invertible" - if self.parent()._charpoly_coefficients.is_in_cache(): + + algebra = self.parent() + if algebra._charpoly_coefficients.is_in_cache(): # We can invert using our charpoly if it will be fast to # compute. If the coefficients are cached, our rank had # better be too! if self.det().is_zero(): raise ZeroDivisionError(not_invertible_msg) - r = self.parent().rank() + r = algebra.rank() a = self.characteristic_polynomial().coefficients(sparse=False) - return (-1)**(r+1)*sum(a[i+1]*self**i for i in range(r))/self.det() + return (-1)**(r+1)*algebra.sum(a[i+1]*self**i + for i in range(r))/self.det() try: inv = (~self.quadratic_representation())(self) @@ -548,7 +555,7 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): If computing my determinant will be fast, we do so and compare with zero (Proposition II.2.4 in Faraut and - Koranyi). Otherwise, Proposition II.3.2 in Faraut and Koranyi + Korányi). Otherwise, Proposition II.3.2 in Faraut and Korányi reduces the problem to the invertibility of my quadratic representation. @@ -560,14 +567,12 @@ 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 +580,6 @@ 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 +662,12 @@ 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 +677,6 @@ 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 +686,6 @@ 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 +742,12 @@ 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 +790,6 @@ 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 +797,6 @@ 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 @@ -816,7 +812,23 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): ALGORITHM: - ......... + First we handle the special cases where the algebra is + trivial, this element is zero, or the dimension of the algebra + is one and this element is not zero. With those out of the + way, we may assume that ``self`` is nonzero, the algebra is + nontrivial, and that the dimension of the algebra is at least + two. + + Beginning with the algebra's unit element (power zero), we add + successive (basis representations of) powers of this element + to a matrix, row-reducing at each step. After row-reducing, we + check the rank of the matrix. If adding a row and row-reducing + does not increase the rank of the matrix at any point, the row + we've just added lives in the span of the previous ones; thus + the corresponding power of ``self`` lives in the span of its + lesser powers. When that happens, the degree of the minimal + polynomial is the rank of the matrix; if it never happens, the + degree must be the dimension of the entire space. SETUP:: @@ -835,7 +847,6 @@ 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 +858,6 @@ 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 @@ -858,11 +868,9 @@ 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 - """ n = self.parent().dimension() @@ -967,7 +975,6 @@ 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 +988,6 @@ 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 @@ -992,9 +998,8 @@ 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: n_max = max(2, JordanSpinEJA._max_random_instance_size()) - sage: n = ZZ.random_element(2, n_max) + sage: d_max = JordanSpinEJA._max_random_instance_dimension() + sage: n = ZZ.random_element(2, max(2,d_max)) sage: J = JordanSpinEJA(n) sage: y = J.random_element() sage: while y == y.coefficient(0)*J.one(): @@ -1009,18 +1014,17 @@ 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) + sage: x = random_eja().random_element() # long time + sage: p = x.minimal_polynomial() # long time + sage: x.apply_univariate_polynomial(p) # long time 0 The minimal polynomial is invariant under a change of basis, and in particular, a re-scaling of the basis:: - sage: set_random_seed() - sage: n_max = RealSymmetricEJA._max_random_instance_size() - sage: n = ZZ.random_element(1, n_max) + sage: d_max = RealSymmetricEJA._max_random_instance_dimension() + 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 +1123,10 @@ 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 +1168,6 @@ 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 @@ -1184,7 +1179,7 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): P = self.parent() left_mult_by_self = lambda y: self*y L = P.module_morphism(function=left_mult_by_self, codomain=P) - return FiniteDimensionalEJAOperator(P, P, L.matrix() ) + return EJAOperator(P, P, L.matrix() ) def quadratic_representation(self, other=None): @@ -1201,7 +1196,6 @@ 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 +1215,6 @@ 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 +1366,7 @@ 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) @@ -1415,9 +1408,8 @@ 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: A = x0.subalgebra_generated_by(orthonormalize=False) sage: x,y,z = A.random_elements(3) sage: (x*y)*z == x*(y*z) True @@ -1425,9 +1417,8 @@ 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.subalgebra_generated_by(orthonormalize=False) sage: A(x^2) == A(x)*A(x) True @@ -1436,7 +1427,6 @@ 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 @@ -1467,8 +1457,7 @@ 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: J = random_eja(field=QQ, orthonormalize=False) sage: x = J.random_element() sage: while x.is_nilpotent() and not J.is_trivial(): ....: x = J.random_element() @@ -1483,7 +1472,10 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): if self.is_nilpotent(): raise ValueError("this only works with non-nilpotent elements!") - J = self.subalgebra_generated_by() + # The subalgebra is transient (we return an element of the + # superalgebra, i.e. this algebra) so why bother + # orthonormalizing? + J = self.subalgebra_generated_by(orthonormalize=False) u = J(self) # The image of the matrix of left-u^m-multiplication @@ -1504,14 +1496,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): # subspace... or do we? Can't we just solve, knowing that # A(c) = u^(s+1) should have a solution in the big space, # too? - # - # Beware, solve_right() means that we're using COLUMN vectors. - # Our FiniteDimensionalAlgebraElement superclass uses rows. u_next = u**(s+1) A = u_next.operator().matrix() c = J.from_vector(A.solve_right(u_next.to_vector())) - # Now c is the idempotent we want, but it still lives in the subalgebra. + # Now c is the idempotent we want, but it still lives in + # the subalgebra. return c.superalgebra_element() @@ -1551,20 +1541,24 @@ 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() sage: (alpha*x + y).trace() == alpha*x.trace() + y.trace() True + The trace of a square is nonnegative:: + + sage: x = random_eja().random_element() + sage: (x*x).trace() >= 0 + True + """ P = self.parent() r = P.rank() @@ -1594,14 +1588,13 @@ 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 sage: x.trace_inner_product(y) == y.trace_inner_product(x) True sage: # bilinear - sage: a = J.base_ring().random_element(); + sage: a = J.base_ring().random_element() sage: actual = (a*(x+z)).trace_inner_product(y) sage: expected = ( a*x.trace_inner_product(y) + ....: a*z.trace_inner_product(y) ) @@ -1648,3 +1641,187 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): """ return self.trace_inner_product(self).sqrt() + + + def operator_trace_inner_product(self, other): + r""" + Return the operator inner product of myself and ``other``. + + The "operator inner product," whose name is not standard, is + defined be the usual linear-algebraic trace of the + ``(x*y).operator()``. + + Proposition III.1.5 in Faraut and Korányi shows that on any + Euclidean Jordan algebra, this is another associative inner + product under which the cone of squares is symmetric. + + This works even if the basis hasn't been orthonormalized + because the eigenvalues of the corresponding matrix don't + change when the basis does (they're preserved by any + similarity transformation). + + SETUP:: + + sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: RealSymmetricEJA, + ....: ComplexHermitianEJA, + ....: random_eja) + + EXAMPLES: + + Proposition III.4.2 of Faraut and Korányi shows that on a + simple algebra of rank `r` and dimension `n`, this inner + product is `n/r` times the canonical + :meth:`trace_inner_product`:: + + sage: J = JordanSpinEJA(4, field=QQ) + sage: x,y = J.random_elements(2) + sage: n = J.dimension() + sage: r = J.rank() + sage: actual = x.operator_trace_inner_product(y) + sage: expected = (n/r)*x.trace_inner_product(y) + sage: actual == expected + True + + :: + + sage: J = RealSymmetricEJA(3) + sage: x,y = J.random_elements(2) + sage: n = J.dimension() + sage: r = J.rank() + sage: actual = x.operator_trace_inner_product(y) + sage: expected = (n/r)*x.trace_inner_product(y) + sage: actual == expected + True + + :: + + sage: J = ComplexHermitianEJA(3, field=QQ, orthonormalize=False) + sage: x,y = J.random_elements(2) + sage: n = J.dimension() + sage: r = J.rank() + sage: actual = x.operator_trace_inner_product(y) + sage: expected = (n/r)*x.trace_inner_product(y) + sage: actual == expected + True + + TESTS: + + The operator inner product is commutative, bilinear, and + associative:: + + sage: J = random_eja() + sage: x,y,z = J.random_elements(3) + sage: # commutative + sage: actual = x.operator_trace_inner_product(y) + sage: expected = y.operator_trace_inner_product(x) + sage: actual == expected + True + sage: # bilinear + sage: a = J.base_ring().random_element() + sage: actual = (a*(x+z)).operator_trace_inner_product(y) + sage: expected = ( a*x.operator_trace_inner_product(y) + + ....: a*z.operator_trace_inner_product(y) ) + sage: actual == expected + True + sage: actual = x.operator_trace_inner_product(a*(y+z)) + sage: expected = ( a*x.operator_trace_inner_product(y) + + ....: a*x.operator_trace_inner_product(z) ) + sage: actual == expected + True + sage: # associative + sage: actual = (x*y).operator_trace_inner_product(z) + sage: expected = y.operator_trace_inner_product(x*z) + sage: actual == expected + True + + Despite the fact that the implementation uses a matrix representation, + the answer is independent of the basis used:: + + sage: J = RealSymmetricEJA(3, field=QQ, orthonormalize=False) + sage: V = RealSymmetricEJA(3) + sage: x,y = J.random_elements(2) + sage: w = V(x.to_matrix()) + sage: z = V(y.to_matrix()) + sage: expected = x.operator_trace_inner_product(y) + sage: actual = w.operator_trace_inner_product(z) + sage: actual == expected + True + + """ + if not other in self.parent(): + raise TypeError("'other' must live in the same algebra") + + return (self*other).operator().matrix().trace() + + + def operator_trace_norm(self): + """ + The norm of this element with respect to + :meth:`operator_trace_inner_product`. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: HadamardEJA) + + EXAMPLES: + + On a simple algebra, this will differ from :meth:`trace_norm` + by the scalar factor ``(n/r).sqrt()``, where `n` is the + dimension of the algebra and `r` its rank. This follows from + the corresponding result (Proposition III.4.2 of Faraut and + Korányi) for the trace inner product:: + + sage: J = HadamardEJA(2) + sage: x = sum(J.gens()) + sage: x.operator_trace_norm() + 1.414213562373095? + + :: + + sage: J = JordanSpinEJA(4) + sage: x = sum(J.gens()) + sage: x.operator_trace_norm() + 4 + + """ + return self.operator_trace_inner_product(self).sqrt() + + +class CartesianProductParentEJAElement(EJAElement): + r""" + An intermediate class for elements that have a Cartesian + product as their parent algebra. + + This is needed because the ``to_matrix`` method (which gives you a + representation from the superalgebra) needs to do special stuff + for Cartesian products. Specifically, an EJA subalgebra of a + Cartesian product EJA will not itself be a Cartesian product (it + has its own basis) -- but we want ``to_matrix()`` to be able to + give us a Cartesian product representation. + """ + 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 in a subclass because the + # _scale() function is slow. + pairs = zip(B, self.to_vector()) + return W.sum( _scale(b, alpha) for (b,alpha) in pairs ) + +class CartesianProductEJAElement(CartesianProductParentEJAElement): + 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() )