mjo/eja/eja_element_subalgebra.py

   1 from sage.matrix.constructor import matrix
   2
   3 from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
   4
   5
   6 class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanSubalgebra):
   7     def __init__(self, elt, orthonormalize_basis):
   8         self._superalgebra = elt.parent()
   9         category = self._superalgebra.category().Associative()
  10         V = self._superalgebra.vector_space()
  11         field = self._superalgebra.base_ring()
  12
  13         # This list is guaranteed to contain all independent powers,
  14         # because it's the maximal set of powers that could possibly
  15         # be independent (by a dimension argument).
  16         powers = [ elt**k for k in range(V.dimension()) ]
  17         power_vectors = [ p.to_vector() for p in powers ]
  18         P = matrix(field, power_vectors)
  19
  20         if orthonormalize_basis == False:
  21             # In this case, we just need to figure out which elements
  22             # of the "powers" list are redundant... First compute the
  23             # vector subspace spanned by the powers of the given
  24             # element.
  25
  26             # Figure out which powers form a linearly-independent set.
  27             ind_rows = P.pivot_rows()
  28
  29             # Pick those out of the list of all powers.
  30             superalgebra_basis = tuple(map(powers.__getitem__, ind_rows))
  31
  32             # If our superalgebra is a subalgebra of something else, then
  33             # these vectors won't have the right coordinates for
  34             # V.span_of_basis() unless we use V.from_vector() on them.
  35             basis_vectors = map(power_vectors.__getitem__, ind_rows)
  36         else:
  37             # If we're going to orthonormalize the basis anyway, we
  38             # might as well just do Gram-Schmidt on the whole list of
  39             # powers. The redundant ones will get zero'd out. If this
  40             # looks like a roundabout way to orthonormalize, it is.
  41             # But converting everything from algebra elements to vectors
  42             # to matrices and then back again turns out to be about
  43             # as fast as reimplementing our own Gram-Schmidt that
  44             # works in an EJA.
  45             G,_ = P.gram_schmidt(orthonormal=True)
  46             basis_vectors = [ g for g in G.rows() if not g.is_zero() ]
  47             superalgebra_basis = [ self._superalgebra.from_vector(b)
  48                                    for b in basis_vectors ]
  49
  50         W = V.span_of_basis( V.from_vector(v) for v in basis_vectors )
  51
  52         fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
  53         fdeja.__init__(self._superalgebra,
  54                        superalgebra_basis,
  55                        category=category)
  56
  57         # The rank is the highest possible degree of a minimal
  58         # polynomial, and is bounded above by the dimension. We know
  59         # in this case that there's an element whose minimal
  60         # polynomial has the same degree as the space's dimension
  61         # (remember how we constructed the space?), so that must be
  62         # its rank too.
  63         self.rank.set_cache(W.dimension())
  64
  65
  66     def _a_regular_element(self):
  67         """
  68         Override the superalgebra method to return the one
  69         regular element that is sure to exist in this
  70         subalgebra, namely the element that generated it.
  71
  72         SETUP::
  73
  74             sage: from mjo.eja.eja_algebra import random_eja
  75
  76         TESTS::
  77
  78             sage: set_random_seed()
  79             sage: J = random_eja().random_element().subalgebra_generated_by()
  80             sage: J._a_regular_element().is_regular()
  81             True
  82
  83         """
  84         if self.dimension() == 0:
  85             return self.zero()
  86         else:
  87             return self.monomial(1)
  88
  89
  90     def one(self):
  91         """
  92         Return the multiplicative identity element of this algebra.
  93
  94         The superclass method computes the identity element, which is
  95         beyond overkill in this case: the superalgebra identity
  96         restricted to this algebra is its identity. Note that we can't
  97         count on the first basis element being the identity -- it migth
  98         have been scaled if we orthonormalized the basis.
  99
 100         SETUP::
 101
 102             sage: from mjo.eja.eja_algebra import (HadamardEJA,
 103             ....:                                  random_eja)
 104
 105         EXAMPLES::
 106
 107             sage: J = HadamardEJA(5)
 108             sage: J.one()
 109             e0 + e1 + e2 + e3 + e4
 110             sage: x = sum(J.gens())
 111             sage: A = x.subalgebra_generated_by()
 112             sage: A.one()
 113             f0
 114             sage: A.one().superalgebra_element()
 115             e0 + e1 + e2 + e3 + e4
 116
 117         TESTS:
 118
 119         The identity element acts like the identity over the rationals::
 120
 121             sage: set_random_seed()
 122             sage: x = random_eja(field=QQ).random_element()
 123             sage: A = x.subalgebra_generated_by()
 124             sage: x = A.random_element()
 125             sage: A.one()*x == x and x*A.one() == x
 126             True
 127
 128         The identity element acts like the identity over the algebraic
 129         reals with an orthonormal basis::
 130
 131             sage: set_random_seed()
 132             sage: x = random_eja().random_element()
 133             sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
 134             sage: x = A.random_element()
 135             sage: A.one()*x == x and x*A.one() == x
 136             True
 137
 138         The matrix of the unit element's operator is the identity over
 139         the rationals::
 140
 141             sage: set_random_seed()
 142             sage: x = random_eja(field=QQ).random_element()
 143             sage: A = x.subalgebra_generated_by()
 144             sage: actual = A.one().operator().matrix()
 145             sage: expected = matrix.identity(A.base_ring(), A.dimension())
 146             sage: actual == expected
 147             True
 148
 149         The matrix of the unit element's operator is the identity over
 150         the algebraic reals with an orthonormal basis::
 151
 152             sage: set_random_seed()
 153             sage: x = random_eja().random_element()
 154             sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
 155             sage: actual = A.one().operator().matrix()
 156             sage: expected = matrix.identity(A.base_ring(), A.dimension())
 157             sage: actual == expected
 158             True
 159
 160         """
 161         if self.dimension() == 0:
 162             return self.zero()
 163         else:
 164             sa_one = self.superalgebra().one().to_vector()
 165             # The extra hackery is because foo.to_vector() might not
 166             # live in foo.parent().vector_space()!
 167             coords = sum( a*b for (a,b)
 168                           in zip(sa_one,
 169                                  self.superalgebra().vector_space().basis()) )
 170             return self.from_vector(self.vector_space().coordinate_vector(coords))
 171