From 98da0ce1d1102057e34646889c10dfa01fa9faec Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sun, 11 Oct 2020 17:24:17 -0400 Subject: [PATCH] eja: make AA the default field because everything cool requires it. --- mjo/eja/eja_algebra.py | 71 +++++++++++++++++-------------- mjo/eja/eja_element.py | 26 +++++------ mjo/eja/eja_element_subalgebra.py | 4 +- mjo/eja/eja_operator.py | 14 +++--- mjo/eja/eja_subalgebra.py | 10 ++--- 5 files changed, 65 insertions(+), 60 deletions(-) diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 1f48f3c..0a26065 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -17,7 +17,7 @@ from sage.misc.lazy_import import lazy_import from sage.misc.prandom import choice from sage.misc.table import table from sage.modules.free_module import FreeModule, VectorSpace -from sage.rings.all import (ZZ, QQ, RR, RLF, CLF, +from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF, PolynomialRing, QuadraticField) from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement @@ -207,8 +207,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Ensure that it says what we think it says:: - sage: JordanSpinEJA(2, field=QQ) - Euclidean Jordan algebra of dimension 2 over Rational Field + sage: JordanSpinEJA(2, field=AA) + Euclidean Jordan algebra of dimension 2 over Algebraic Real Field sage: JordanSpinEJA(3, field=RDF) Euclidean Jordan algebra of dimension 3 over Real Double Field @@ -551,8 +551,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Finite family {0: e0, 1: e1, 2: e2} sage: J.natural_basis() ( - [1 0] [ 0 1/2*sqrt2] [0 0] - [0 0], [1/2*sqrt2 0], [0 1] + [1 0] [ 0 0.7071067811865475?] [0 0] + [0 0], [0.7071067811865475? 0], [0 1] ) :: @@ -1013,7 +1013,7 @@ class KnownRankEJA(object): return 5 @classmethod - def random_instance(cls, field=QQ, **kwargs): + def random_instance(cls, field=AA, **kwargs): """ Return a random instance of this type of algebra. @@ -1069,7 +1069,7 @@ class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): (r0, r1, r2) """ - def __init__(self, n, field=QQ, **kwargs): + def __init__(self, n, field=AA, **kwargs): V = VectorSpace(field, n) mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ] for i in range(n) ] @@ -1102,7 +1102,7 @@ class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): return x.to_vector().inner_product(y.to_vector()) -def random_eja(field=QQ, nontrivial=False): +def random_eja(field=AA, nontrivial=False): """ Return a "random" finite-dimensional Euclidean Jordan Algebra. @@ -1322,8 +1322,8 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): In theory, our "field" can be any subfield of the reals:: - sage: RealSymmetricEJA(2, AA) - Euclidean Jordan algebra of dimension 3 over Algebraic Real Field + sage: RealSymmetricEJA(2, RDF) + Euclidean Jordan algebra of dimension 3 over Real Double Field sage: RealSymmetricEJA(2, RR) Euclidean Jordan algebra of dimension 3 over Real Field with 53 bits of precision @@ -1415,7 +1415,7 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): return 4 # Dimension 10 - def __init__(self, n, field=QQ, **kwargs): + def __init__(self, n, field=AA, **kwargs): basis = self._denormalized_basis(n, field) super(RealSymmetricEJA, self).__init__(field, basis, n, **kwargs) @@ -1435,7 +1435,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): EXAMPLES:: - sage: F = QuadraticField(-1, 'i') + sage: F = QuadraticField(-1, 'I') sage: x1 = F(4 - 2*i) sage: x2 = F(1 + 2*i) sage: x3 = F(-i) @@ -1455,7 +1455,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): sage: set_random_seed() sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size() sage: n = ZZ.random_element(n_max) - sage: F = QuadraticField(-1, 'i') + sage: F = QuadraticField(-1, 'I') sage: X = random_matrix(F, n) sage: Y = random_matrix(F, n) sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X) @@ -1498,15 +1498,15 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): ....: [ 9, 10, 11, 12], ....: [-10, 9, -12, 11] ]) sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A) - [ 2*i + 1 4*i + 3] - [ 10*i + 9 12*i + 11] + [ 2*I + 1 4*I + 3] + [ 10*I + 9 12*I + 11] TESTS: Unembedding is the inverse of embedding:: sage: set_random_seed() - sage: F = QuadraticField(-1, 'i') + sage: F = QuadraticField(-1, 'I') sage: M = random_matrix(F, 3) sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M) sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M @@ -1524,7 +1524,12 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): field = M.base_ring() R = PolynomialRing(field, 'z') z = R.gen() - F = field.extension(z**2 + 1, 'i', embedding=CLF(-1).sqrt()) + if field is AA: + # Sage doesn't know how to embed AA into QQbar, i.e. how + # to adjoin sqrt(-1) to AA. + F = QQbar + else: + F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt()) i = F.gen() # Go top-left to bottom-right (reading order), converting every @@ -1565,7 +1570,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): sage: Ye = y.natural_representation() sage: X = ComplexHermitianEJA.real_unembed(Xe) sage: Y = ComplexHermitianEJA.real_unembed(Ye) - sage: expected = (X*Y).trace().vector()[0] + sage: expected = (X*Y).trace().real() sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye) sage: actual == expected True @@ -1589,8 +1594,8 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): In theory, our "field" can be any subfield of the reals:: - sage: ComplexHermitianEJA(2, AA) - Euclidean Jordan algebra of dimension 4 over Algebraic Real Field + sage: ComplexHermitianEJA(2, RDF) + Euclidean Jordan algebra of dimension 4 over Real Double Field sage: ComplexHermitianEJA(2, RR) Euclidean Jordan algebra of dimension 4 over Real Field with 53 bits of precision @@ -1700,7 +1705,7 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): return ( s.change_ring(field) for s in S ) - def __init__(self, n, field=QQ, **kwargs): + def __init__(self, n, field=AA, **kwargs): basis = self._denormalized_basis(n,field) super(ComplexHermitianEJA,self).__init__(field, basis, n, **kwargs) @@ -1752,7 +1757,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): if M.ncols() != n: raise ValueError("the matrix 'M' must be square") - F = QuadraticField(-1, 'i') + F = QuadraticField(-1, 'I') i = F.gen() blocks = [] @@ -1828,10 +1833,10 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0].conjugate(): raise ValueError('bad off-diagonal submatrix') - z = submat[0,0].vector()[0] # real part - z += submat[0,0].vector()[1]*i # imag part - z += submat[0,1].vector()[0]*j # real part - z += submat[0,1].vector()[1]*k # imag part + z = submat[0,0].real() + z += submat[0,0].imag()*i + z += submat[0,1].real()*j + z += submat[0,1].imag()*k elements.append(z) return matrix(Q, n/4, elements) @@ -1884,8 +1889,8 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, In theory, our "field" can be any subfield of the reals:: - sage: QuaternionHermitianEJA(2, AA) - Euclidean Jordan algebra of dimension 6 over Algebraic Real Field + sage: QuaternionHermitianEJA(2, RDF) + Euclidean Jordan algebra of dimension 6 over Real Double Field sage: QuaternionHermitianEJA(2, RR) Euclidean Jordan algebra of dimension 6 over Real Field with 53 bits of precision @@ -1996,7 +2001,7 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, return ( s.change_ring(field) for s in S ) - def __init__(self, n, field=QQ, **kwargs): + def __init__(self, n, field=AA, **kwargs): basis = self._denormalized_basis(n,field) super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs) @@ -2056,7 +2061,7 @@ class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): sage: actual == expected True """ - def __init__(self, n, field=QQ, B=None, **kwargs): + def __init__(self, n, field=AA, B=None, **kwargs): if B is None: self._B = matrix.identity(field, max(0,n-1)) else: @@ -2169,7 +2174,7 @@ class JordanSpinEJA(BilinearFormEJA): True """ - def __init__(self, n, field=QQ, **kwargs): + def __init__(self, n, field=AA, **kwargs): # This is a special case of the BilinearFormEJA with the identity # matrix as its bilinear form. return super(JordanSpinEJA, self).__init__(n, field, **kwargs) @@ -2199,12 +2204,12 @@ class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): sage: J.one().norm() 0 sage: J.one().subalgebra_generated_by() - Euclidean Jordan algebra of dimension 0 over Rational Field + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field sage: J.rank() 0 """ - def __init__(self, field=QQ, **kwargs): + def __init__(self, field=AA, **kwargs): mult_table = [] fdeja = super(TrivialEJA, self) # The rank is zero using my definition, namely the dimension of the diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index 7c4c79d..0f6a47c 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -113,7 +113,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): We should always get back an element of the algebra:: sage: set_random_seed() - sage: p = PolynomialRing(QQ, 't').random_element() + sage: p = PolynomialRing(AA, 't').random_element() sage: J = random_eja() sage: x = J.random_element() sage: x.apply_univariate_polynomial(p) in J @@ -575,7 +575,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): The spectral decomposition of a non-regular element should always contain at least one non-minimal idempotent:: - sage: J = RealSymmetricEJA(3, AA) + sage: J = RealSymmetricEJA(3) sage: x = sum(J.gens()) sage: x.is_regular() False @@ -586,7 +586,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): On the other hand, the spectral decomposition of a regular element should always be in terms of minimal idempotents:: - sage: J = JordanSpinEJA(4, AA) + sage: J = JordanSpinEJA(4) sage: x = sum( i*J.gens()[i] for i in range(len(J.gens())) ) sage: x.is_regular() True @@ -909,9 +909,9 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: set_random_seed() sage: n_max = RealSymmetricEJA._max_test_case_size() sage: n = ZZ.random_element(1, n_max) - sage: J1 = RealSymmetricEJA(n,QQ) - sage: J2 = RealSymmetricEJA(n,QQ,normalize_basis=False) - sage: X = random_matrix(QQ,n) + sage: J1 = RealSymmetricEJA(n) + sage: J2 = RealSymmetricEJA(n,normalize_basis=False) + sage: X = random_matrix(AA,n) sage: X = X*X.transpose() sage: x1 = J1(X) sage: x2 = J2(X) @@ -1003,7 +1003,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: J = HadamardEJA(2) sage: x = sum(J.gens()) sage: x.norm() - sqrt(2) + 1.414213562373095? :: @@ -1065,10 +1065,10 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: n = x_vec.degree() sage: x0 = x_vec[0] sage: x_bar = x_vec[1:] - sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)]) + sage: A = matrix(AA, 1, [x_vec.inner_product(x_vec)]) sage: B = 2*x0*x_bar.row() sage: C = 2*x0*x_bar.column() - sage: D = matrix.identity(QQ, n-1) + sage: D = matrix.identity(AA, 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 = matrix.block(2,2,[A,B,C,D]) @@ -1192,7 +1192,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): The spectral decomposition of the identity is ``1`` times itself, and the spectral decomposition of zero is ``0`` times the identity:: - sage: J = RealSymmetricEJA(3,AA) + sage: J = RealSymmetricEJA(3) sage: J.one() e0 + e2 + e5 sage: J.one().spectral_decomposition() @@ -1202,7 +1202,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): TESTS:: - sage: J = RealSymmetricEJA(4,AA) + sage: J = RealSymmetricEJA(4) sage: x = sum(J.gens()) sage: sd = x.spectral_decomposition() sage: l0 = sd[0][0] @@ -1453,14 +1453,14 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: J = HadamardEJA(2) sage: x = sum(J.gens()) sage: x.trace_norm() - sqrt(2) + 1.414213562373095? :: sage: J = JordanSpinEJA(4) sage: x = sum(J.gens()) sage: x.trace_norm() - 2*sqrt(2) + 2.828427124746190? """ return self.trace_inner_product(self).sqrt() diff --git a/mjo/eja/eja_element_subalgebra.py b/mjo/eja/eja_element_subalgebra.py index 292871a..67342c6 100644 --- a/mjo/eja/eja_element_subalgebra.py +++ b/mjo/eja/eja_element_subalgebra.py @@ -130,7 +130,7 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide reals with an orthonormal basis:: sage: set_random_seed() - sage: x = random_eja(AA).random_element() + sage: x = random_eja().random_element() sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) sage: x = A.random_element() sage: A.one()*x == x and x*A.one() == x @@ -151,7 +151,7 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide the algebraic reals with an orthonormal basis:: sage: set_random_seed() - sage: x = random_eja(AA).random_element() + sage: x = random_eja().random_element() sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) sage: actual = A.one().operator().matrix() sage: expected = matrix.identity(A.base_ring(), A.dimension()) diff --git a/mjo/eja/eja_operator.py b/mjo/eja/eja_operator.py index ee33dbf..667e3d5 100644 --- a/mjo/eja/eja_operator.py +++ b/mjo/eja/eja_operator.py @@ -125,9 +125,9 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): sage: J1 = JordanSpinEJA(3) sage: J2 = HadamardEJA(2) sage: J3 = RealSymmetricEJA(1) - sage: mat1 = matrix(QQ, [[1,2,3], + sage: mat1 = matrix(AA, [[1,2,3], ....: [4,5,6]]) - sage: mat2 = matrix(QQ, [[7,8]]) + sage: mat2 = matrix(AA, [[7,8]]) sage: g = FiniteDimensionalEuclideanJordanAlgebraOperator(J1, ....: J2, ....: mat1) @@ -139,9 +139,9 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): algebras represented by the matrix: [39 54 69] Domain: Euclidean Jordan algebra of dimension 3 over - Rational Field + Algebraic Real Field Codomain: Euclidean Jordan algebra of dimension 1 over - Rational Field + Algebraic Real Field """ return FiniteDimensionalEuclideanJordanAlgebraOperator( @@ -341,9 +341,9 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): [1 0] [0 1] Domain: Euclidean Jordan algebra of dimension 2 over - Rational Field + Algebraic Real Field Codomain: Euclidean Jordan algebra of dimension 2 over - Rational Field + Algebraic Real Field """ msg = ("Linear operator between finite-dimensional Euclidean Jordan " @@ -542,7 +542,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): EXAMPLES:: - sage: J = RealSymmetricEJA(4,AA) + sage: J = RealSymmetricEJA(4) sage: x = sum(J.gens()) sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) sage: L0x = A(x).operator() diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 0be8561..ac77f22 100644 --- a/mjo/eja/eja_subalgebra.py +++ b/mjo/eja/eja_subalgebra.py @@ -95,9 +95,9 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda matrices do not contain the superalgebra's identity element:: sage: J = RealSymmetricEJA(2) - sage: E11 = matrix(QQ, [ [1,0], + sage: E11 = matrix(AA, [ [1,0], ....: [0,0] ]) - sage: E22 = matrix(QQ, [ [0,0], + sage: E22 = matrix(AA, [ [0,0], ....: [0,1] ]) sage: K1 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E11),)) sage: K1.one().natural_representation() @@ -198,7 +198,7 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda EXAMPLES:: sage: J = RealSymmetricEJA(3) - sage: X = matrix(QQ, [ [0,0,1], + sage: X = matrix(AA, [ [0,0,1], ....: [0,1,0], ....: [1,0,0] ]) sage: x = J(X) @@ -249,10 +249,10 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda EXAMPLES:: sage: J = RealSymmetricEJA(3) - sage: E11 = matrix(QQ, [ [1,0,0], + sage: E11 = matrix(ZZ, [ [1,0,0], ....: [0,0,0], ....: [0,0,0] ]) - sage: E22 = matrix(QQ, [ [0,0,0], + sage: E22 = matrix(ZZ, [ [0,0,0], ....: [0,1,0], ....: [0,0,0] ]) sage: b1 = J(E11) -- 2.44.2