mjo/eja/euclidean_jordan_algebra.py

   1 """
   2 Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
   3 specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
   4 are used in optimization, and have some additional nice methods beyond
   5 what can be supported in a general Jordan Algebra.
   6 """
   7
   8 from sage.all import *
   9
  10 def eja_minimal_polynomial(x):
  11     """
  12     Return the minimal polynomial of ``x`` in its parent EJA
  13     """
  14     return x._x.matrix().minimal_polynomial()
  15
  16
  17 def eja_rn(dimension, field=QQ):
  18     """
  19     Return the Euclidean Jordan Algebra corresponding to the set
  20     `R^n` under the Hadamard product.
  21
  22     EXAMPLES:
  23
  24     This multiplication table can be verified by hand::
  25
  26         sage: J = eja_rn(3)
  27         sage: e0,e1,e2 = J.gens()
  28         sage: e0*e0
  29         e0
  30         sage: e0*e1
  31         0
  32         sage: e0*e2
  33         0
  34         sage: e1*e1
  35         e1
  36         sage: e1*e2
  37         0
  38         sage: e2*e2
  39         e2
  40
  41     """
  42     # The FiniteDimensionalAlgebra constructor takes a list of
  43     # matrices, the ith representing right multiplication by the ith
  44     # basis element in the vector space. So if e_1 = (1,0,0), then
  45     # right (Hadamard) multiplication of x by e_1 picks out the first
  46     # component of x; and likewise for the ith basis element e_i.
  47     Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
  48            for i in xrange(dimension) ]
  49     A = FiniteDimensionalAlgebra(QQ,Qs,assume_associative=True)
  50     return JordanAlgebra(A)
  51
  52
  53 def eja_ln(dimension, field=QQ):
  54     """
  55     Return the Jordan algebra corresponding to the Lorenzt "ice cream"
  56     cone of the given ``dimension``.
  57
  58     EXAMPLES:
  59
  60     This multiplication table can be verified by hand::
  61
  62         sage: J = eja_ln(4)
  63         sage: e0,e1,e2,e3 = J.gens()
  64         sage: e0*e0
  65         e0
  66         sage: e0*e1
  67         e1
  68         sage: e0*e2
  69         e2
  70         sage: e0*e3
  71         e3
  72         sage: e1*e2
  73         0
  74         sage: e1*e3
  75         0
  76         sage: e2*e3
  77         0
  78
  79     In one dimension, this is the reals under multiplication::
  80
  81       sage: J1 = eja_ln(1)
  82       sage: J2 = eja_rn(1)
  83       sage: J1 == J2
  84       True
  85
  86     """
  87     Qs = []
  88     id_matrix = identity_matrix(field,dimension)
  89     for i in xrange(dimension):
  90         ei = id_matrix.column(i)
  91         Qi = zero_matrix(field,dimension)
  92         Qi.set_row(0, ei)
  93         Qi.set_column(0, ei)
  94         Qi += diagonal_matrix(dimension, [ei[0]]*dimension)
  95         # The addition of the diagonal matrix adds an extra ei[0] in the
  96         # upper-left corner of the matrix.
  97         Qi[0,0] = Qi[0,0] * ~field(2)
  98         Qs.append(Qi)
  99
 100     A = FiniteDimensionalAlgebra(QQ,Qs,assume_associative=True)
 101     return JordanAlgebra(A)