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             basis_vectors = map(power_vectors.__getitem__, ind_rows)
  32         else:
  33             # If we're going to orthonormalize the basis anyway, we
  34             # might as well just do Gram-Schmidt on the whole list of
  35             # powers. The redundant ones will get zero'd out. If this
  36             # looks like a roundabout way to orthonormalize, it is.
  37             # But converting everything from algebra elements to vectors
  38             # to matrices and then back again turns out to be about
  39             # as fast as reimplementing our own Gram-Schmidt that
  40             # works in an EJA.
  41             G,_ = P.gram_schmidt(orthonormal=True)
  42             basis_vectors = [ g for g in G.rows() if not g.is_zero() ]
  43             superalgebra_basis = [ self._superalgebra.from_vector(b)
  44                                    for b in basis_vectors ]
  45
  46         fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
  47         fdeja.__init__(self._superalgebra,
  48                        superalgebra_basis,
  49                        category=category,
  50                        check_axioms=False)
  51
  52         # The rank is the highest possible degree of a minimal
  53         # polynomial, and is bounded above by the dimension. We know
  54         # in this case that there's an element whose minimal
  55         # polynomial has the same degree as the space's dimension
  56         # (remember how we constructed the space?), so that must be
  57         # its rank too.
  58         self.rank.set_cache(self.dimension())
  59
  60
  61     def one(self):
  62         """
  63         Return the multiplicative identity element of this algebra.
  64
  65         The superclass method computes the identity element, which is
  66         beyond overkill in this case: the superalgebra identity
  67         restricted to this algebra is its identity. Note that we can't
  68         count on the first basis element being the identity -- it migth
  69         have been scaled if we orthonormalized the basis.
  70
  71         SETUP::
  72
  73             sage: from mjo.eja.eja_algebra import (HadamardEJA,
  74             ....:                                  random_eja)
  75
  76         EXAMPLES::
  77
  78             sage: J = HadamardEJA(5)
  79             sage: J.one()
  80             e0 + e1 + e2 + e3 + e4
  81             sage: x = sum(J.gens())
  82             sage: A = x.subalgebra_generated_by()
  83             sage: A.one()
  84             f0
  85             sage: A.one().superalgebra_element()
  86             e0 + e1 + e2 + e3 + e4
  87
  88         TESTS:
  89
  90         The identity element acts like the identity over the rationals::
  91
  92             sage: set_random_seed()
  93             sage: x = random_eja(field=QQ).random_element()
  94             sage: A = x.subalgebra_generated_by()
  95             sage: x = A.random_element()
  96             sage: A.one()*x == x and x*A.one() == x
  97             True
  98
  99         The identity element acts like the identity over the algebraic
 100         reals with an orthonormal basis::
 101
 102             sage: set_random_seed()
 103             sage: x = random_eja().random_element()
 104             sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
 105             sage: x = A.random_element()
 106             sage: A.one()*x == x and x*A.one() == x
 107             True
 108
 109         The matrix of the unit element's operator is the identity over
 110         the rationals::
 111
 112             sage: set_random_seed()
 113             sage: x = random_eja(field=QQ).random_element()
 114             sage: A = x.subalgebra_generated_by()
 115             sage: actual = A.one().operator().matrix()
 116             sage: expected = matrix.identity(A.base_ring(), A.dimension())
 117             sage: actual == expected
 118             True
 119
 120         The matrix of the unit element's operator is the identity over
 121         the algebraic reals with an orthonormal basis::
 122
 123             sage: set_random_seed()
 124             sage: x = random_eja().random_element()
 125             sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
 126             sage: actual = A.one().operator().matrix()
 127             sage: expected = matrix.identity(A.base_ring(), A.dimension())
 128             sage: actual == expected
 129             True
 130
 131         """
 132         if self.dimension() == 0:
 133             return self.zero()
 134         else:
 135             sa_one = self.superalgebra().one().to_vector()
 136             # The extra hackery is because foo.to_vector() might not
 137             # live in foo.parent().vector_space()!
 138             coords = sum( a*b for (a,b)
 139                           in zip(sa_one,
 140                                  self.superalgebra().vector_space().basis()) )
 141             return self.from_vector(self.vector_space().coordinate_vector(coords))
 142