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                        check_axioms=False)
  57
  58         # The rank is the highest possible degree of a minimal
  59         # polynomial, and is bounded above by the dimension. We know
  60         # in this case that there's an element whose minimal
  61         # polynomial has the same degree as the space's dimension
  62         # (remember how we constructed the space?), so that must be
  63         # its rank too.
  64         self.rank.set_cache(W.dimension())
  65
  66
  67     def one(self):
  68         """
  69         Return the multiplicative identity element of this algebra.
  70
  71         The superclass method computes the identity element, which is
  72         beyond overkill in this case: the superalgebra identity
  73         restricted to this algebra is its identity. Note that we can't
  74         count on the first basis element being the identity -- it migth
  75         have been scaled if we orthonormalized the basis.
  76
  77         SETUP::
  78
  79             sage: from mjo.eja.eja_algebra import (HadamardEJA,
  80             ....:                                  random_eja)
  81
  82         EXAMPLES::
  83
  84             sage: J = HadamardEJA(5)
  85             sage: J.one()
  86             e0 + e1 + e2 + e3 + e4
  87             sage: x = sum(J.gens())
  88             sage: A = x.subalgebra_generated_by()
  89             sage: A.one()
  90             f0
  91             sage: A.one().superalgebra_element()
  92             e0 + e1 + e2 + e3 + e4
  93
  94         TESTS:
  95
  96         The identity element acts like the identity over the rationals::
  97
  98             sage: set_random_seed()
  99             sage: x = random_eja(field=QQ).random_element()
 100             sage: A = x.subalgebra_generated_by()
 101             sage: x = A.random_element()
 102             sage: A.one()*x == x and x*A.one() == x
 103             True
 104
 105         The identity element acts like the identity over the algebraic
 106         reals with an orthonormal basis::
 107
 108             sage: set_random_seed()
 109             sage: x = random_eja().random_element()
 110             sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
 111             sage: x = A.random_element()
 112             sage: A.one()*x == x and x*A.one() == x
 113             True
 114
 115         The matrix of the unit element's operator is the identity over
 116         the rationals::
 117
 118             sage: set_random_seed()
 119             sage: x = random_eja(field=QQ).random_element()
 120             sage: A = x.subalgebra_generated_by()
 121             sage: actual = A.one().operator().matrix()
 122             sage: expected = matrix.identity(A.base_ring(), A.dimension())
 123             sage: actual == expected
 124             True
 125
 126         The matrix of the unit element's operator is the identity over
 127         the algebraic reals with an orthonormal basis::
 128
 129             sage: set_random_seed()
 130             sage: x = random_eja().random_element()
 131             sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
 132             sage: actual = A.one().operator().matrix()
 133             sage: expected = matrix.identity(A.base_ring(), A.dimension())
 134             sage: actual == expected
 135             True
 136
 137         """
 138         if self.dimension() == 0:
 139             return self.zero()
 140         else:
 141             sa_one = self.superalgebra().one().to_vector()
 142             # The extra hackery is because foo.to_vector() might not
 143             # live in foo.parent().vector_space()!
 144             coords = sum( a*b for (a,b)
 145                           in zip(sa_one,
 146                                  self.superalgebra().vector_space().basis()) )
 147             return self.from_vector(self.vector_space().coordinate_vector(coords))
 148