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):
"""
SETUP::
- sage: from mjo.eja.eja_algebra import HadamardEJA
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: HadamardEJA)
EXAMPLES:
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::
sage: J = HadamardEJA(3)
sage: p1 = J.one().characteristic_polynomial()
sage: q1 = J.zero().characteristic_polynomial()
- sage: e0,e1,e2 = J.gens()
- sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3
+ sage: b0,b1,b2 = J.gens()
+ sage: A = (b0 + 2*b1 + 3*b2).subalgebra_generated_by() # dim 3
sage: p2 = A.one().characteristic_polynomial()
sage: q2 = A.zero().characteristic_polynomial()
sage: p1 == p2
EXAMPLES::
sage: J = JordanSpinEJA(2)
- sage: e0,e1 = J.gens()
sage: x = sum( J.gens() )
sage: x.det()
0
::
sage: J = JordanSpinEJA(3)
- sage: e0,e1,e2 = J.gens()
sage: x = sum( J.gens() )
sage: x.det()
-1
sage: (x*y).det() == x.det()*y.det()
True
- The determinant in matrix algebras is just the usual determinant::
+ The determinant in real matrix algebras is the usual determinant::
sage: set_random_seed()
sage: X = matrix.random(QQ,3)
sage: actual2 == expected
True
- ::
-
- sage: set_random_seed()
- sage: J1 = ComplexHermitianEJA(2)
- sage: J2 = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
- sage: X = matrix.random(GaussianIntegers(), 2)
- sage: X = X + X.H
- sage: expected = AA(X.det())
- sage: actual1 = J1(J1.real_embed(X)).det()
- sage: actual2 = J2(J2.real_embed(X)).det()
- sage: expected == actual1
- True
- sage: expected == actual2
- True
-
"""
P = self.parent()
r = P.rank()
sage: J = JordanSpinEJA(5)
sage: J.one().is_regular()
False
- sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
+ sage: b0, b1, b2, b3, b4 = J.gens()
+ sage: b0 == J.one()
+ True
sage: for x in J.gens():
....: (J.one() + x).is_regular()
False
sage: J = JordanSpinEJA(4)
sage: J.one().degree()
1
- sage: e0,e1,e2,e3 = J.gens()
- sage: (e0 - e1).degree()
+ sage: b0,b1,b2,b3 = J.gens()
+ sage: (b0 - b1).degree()
2
In the spin factor algebra (of rank two), all elements that
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():
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)
sage: J = ComplexHermitianEJA(3)
sage: J.one()
- e0 + e3 + e8
+ b0 + b3 + b8
sage: J.one().to_matrix()
- [1 0 0 0 0 0]
- [0 1 0 0 0 0]
- [0 0 1 0 0 0]
- [0 0 0 1 0 0]
- [0 0 0 0 1 0]
- [0 0 0 0 0 1]
+ +---+---+---+
+ | 1 | 0 | 0 |
+ +---+---+---+
+ | 0 | 1 | 0 |
+ +---+---+---+
+ | 0 | 0 | 1 |
+ +---+---+---+
::
sage: J = QuaternionHermitianEJA(2)
sage: J.one()
- e0 + e5
+ b0 + b5
sage: J.one().to_matrix()
- [1 0 0 0 0 0 0 0]
- [0 1 0 0 0 0 0 0]
- [0 0 1 0 0 0 0 0]
- [0 0 0 1 0 0 0 0]
- [0 0 0 0 1 0 0 0]
- [0 0 0 0 0 1 0 0]
- [0 0 0 0 0 0 1 0]
- [0 0 0 0 0 0 0 1]
+ +---+---+
+ | 1 | 0 |
+ +---+---+
+ | 0 | 1 |
+ +---+---+
This also works in Cartesian product algebras::
sage: J = RealSymmetricEJA(3)
sage: J.one()
- e0 + e2 + e5
+ b0 + b2 + b5
sage: J.one().spectral_decomposition()
- [(1, e0 + e2 + e5)]
+ [(1, b0 + b2 + b5)]
sage: J.zero().spectral_decomposition()
- [(0, e0 + e2 + e5)]
+ [(0, b0 + b2 + b5)]
TESTS::
The spectral decomposition should work in subalgebras, too::
sage: J = RealSymmetricEJA(4)
- sage: (e0, e1, e2, e3, e4, e5, e6, e7, e8, e9) = J.gens()
- sage: A = 2*e5 - 2*e8
+ sage: (b0, b1, b2, b3, b4, b5, b6, b7, b8, b9) = J.gens()
+ sage: A = 2*b5 - 2*b8
sage: (lambda1, c1) = A.spectral_decomposition()[1]
sage: (J0, J5, J1) = J.peirce_decomposition(c1)
sage: (f0, f1, f2) = J1.gens()
sage: f0.spectral_decomposition()
- [(0, f2), (1, f0)]
+ [(0, 1.000000000000000?*c2), (1, 1.000000000000000?*c0)]
"""
A = self.subalgebra_generated_by(orthonormalize=True)
projects / sage.d.git / blobdiff