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.categories.magmatic_algebras import MagmaticAlgebras
   9 from sage.structure.element import is_Matrix
  10 from sage.structure.category_object import normalize_names
  11
  12 from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra
  13 from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement
  14 from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_morphism import FiniteDimensionalAlgebraMorphism
  15
  16
  17 class FiniteDimensionalEuclideanJordanAlgebraMorphism(FiniteDimensionalAlgebraMorphism):
  18     """
  19     A very thin wrapper around FiniteDimensionalAlgebraMorphism that
  20     does only three things:
  21
  22       1. Avoids the ``unitary`` and ``check`` arguments to the constructor
  23          that will always be ``False``. This is necessary because these
  24          are homomorphisms with respect to ADDITION, but the SageMath
  25          machinery wants to check that they're homomorphisms with respect
  26          to (Jordan) MULTIPLICATION. That obviously doesn't work.
  27
  28       2. Inputs and outputs the underlying matrix with respect to COLUMN
  29          vectors, unlike the parent class.
  30
  31       3. Allows us to add morphisms in the obvious way.
  32
  33     If this seems a bit heavyweight, it is. I would have been happy to
  34     use a the ring morphism that underlies the finite-dimensional
  35     algebra morphism, but they don't seem to be callable on elements of
  36     our EJA, and you can't add them.
  37     """
  38
  39     def __add__(self, other):
  40         """
  41         Add two EJA morphisms in the obvious way.
  42
  43         EXAMPLES:
  44
  45             sage: J = RealSymmetricEJA(3)
  46             sage: x = J.zero()
  47             sage: y = J.zero()
  48             sage: x.operator() + y.operator()
  49             Morphism from Euclidean Jordan algebra of degree 6 over Rational
  50             Field to Euclidean Jordan algebra of degree 6 over Rational Field
  51             given by matrix
  52             [0 0 0 0 0 0]
  53             [0 0 0 0 0 0]
  54             [0 0 0 0 0 0]
  55             [0 0 0 0 0 0]
  56             [0 0 0 0 0 0]
  57             [0 0 0 0 0 0]
  58
  59         """
  60         P = self.parent()
  61         if not other in P:
  62             raise ValueError("summands must live in the same space")
  63
  64         return FiniteDimensionalEuclideanJordanAlgebraMorphism(
  65                   P,
  66                   self.matrix() + other.matrix() )
  67
  68
  69     def __init__(self, parent, f):
  70         FiniteDimensionalAlgebraMorphism.__init__(self,
  71                                                   parent,
  72                                                   f.transpose(),
  73                                                   unitary=False,
  74                                                   check=False)
  75
  76
  77     def _repr_(self):
  78         """
  79         We override only the representation that is shown to the user,
  80         because we want the matrix to be with respect to COLUMN vectors.
  81
  82         TESTS:
  83
  84         Ensure that we see the transpose of the underlying matrix object:
  85
  86             sage: J = RealSymmetricEJA(3)
  87             sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens()))
  88             sage: L = x.operator()
  89             sage: L
  90             Morphism from Euclidean Jordan algebra of degree 6 over Rational
  91             Field to Euclidean Jordan algebra of degree 6 over Rational Field
  92             given by matrix
  93             [  0   1   2   0   0   0]
  94             [1/2 3/2   2 1/2   1   0]
  95             [  1   2 5/2   0 1/2   1]
  96             [  0   1   0   3   4   0]
  97             [  0   1 1/2   2   4   2]
  98             [  0   0   2   0   4   5]
  99             sage: L._matrix
 100             [  0 1/2   1   0   0   0]
 101             [  1 3/2   2   1   1   0]
 102             [  2   2 5/2   0 1/2   2]
 103             [  0 1/2   0   3   2   0]
 104             [  0   1 1/2   4   4   4]
 105             [  0   0   1   0   2   5]
 106
 107         """
 108         return "Morphism from {} to {} given by matrix\n{}".format(
 109             self.domain(), self.codomain(), self.matrix())
 110
 111     def matrix(self):
 112         """
 113         Return the matrix of this morphism with respect to a left-action
 114         on column vectors.
 115         """
 116         return FiniteDimensionalAlgebraMorphism.matrix(self).transpose()
 117
 118
 119 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
 120     @staticmethod
 121     def __classcall_private__(cls,
 122                               field,
 123                               mult_table,
 124                               names='e',
 125                               assume_associative=False,
 126                               category=None,
 127                               rank=None,
 128                               natural_basis=None):
 129         n = len(mult_table)
 130         mult_table = [b.base_extend(field) for b in mult_table]
 131         for b in mult_table:
 132             b.set_immutable()
 133             if not (is_Matrix(b) and b.dimensions() == (n, n)):
 134                 raise ValueError("input is not a multiplication table")
 135         mult_table = tuple(mult_table)
 136
 137         cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis()
 138         cat.or_subcategory(category)
 139         if assume_associative:
 140             cat = cat.Associative()
 141
 142         names = normalize_names(n, names)
 143
 144         fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls)
 145         return fda.__classcall__(cls,
 146                                  field,
 147                                  mult_table,
 148                                  assume_associative=assume_associative,
 149                                  names=names,
 150                                  category=cat,
 151                                  rank=rank,
 152                                  natural_basis=natural_basis)
 153
 154
 155     def __init__(self,
 156                  field,
 157                  mult_table,
 158                  names='e',
 159                  assume_associative=False,
 160                  category=None,
 161                  rank=None,
 162                  natural_basis=None):
 163         """
 164         EXAMPLES:
 165
 166         By definition, Jordan multiplication commutes::
 167
 168             sage: set_random_seed()
 169             sage: J = random_eja()
 170             sage: x = J.random_element()
 171             sage: y = J.random_element()
 172             sage: x*y == y*x
 173             True
 174
 175         """
 176         self._rank = rank
 177         self._natural_basis = natural_basis
 178         self._multiplication_table = mult_table
 179         fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
 180         fda.__init__(field,
 181                      mult_table,
 182                      names=names,
 183                      category=category)
 184
 185
 186     def _repr_(self):
 187         """
 188         Return a string representation of ``self``.
 189         """
 190         fmt = "Euclidean Jordan algebra of degree {} over {}"
 191         return fmt.format(self.degree(), self.base_ring())
 192
 193
 194     def _a_regular_element(self):
 195         """
 196         Guess a regular element. Needed to compute the basis for our
 197         characteristic polynomial coefficients.
 198         """
 199         gs = self.gens()
 200         z = self.sum( (i+1)*gs[i] for i in range(len(gs)) )
 201         if not z.is_regular():
 202             raise ValueError("don't know a regular element")
 203         return z
 204
 205
 206     @cached_method
 207     def _charpoly_basis_space(self):
 208         """
 209         Return the vector space spanned by the basis used in our
 210         characteristic polynomial coefficients. This is used not only to
 211         compute those coefficients, but also any time we need to
 212         evaluate the coefficients (like when we compute the trace or
 213         determinant).
 214         """
 215         z = self._a_regular_element()
 216         V = z.vector().parent().ambient_vector_space()
 217         V1 = V.span_of_basis( (z**k).vector() for k in range(self.rank()) )
 218         b =  (V1.basis() + V1.complement().basis())
 219         return V.span_of_basis(b)
 220
 221
 222     @cached_method
 223     def _charpoly_coeff(self, i):
 224         """
 225         Return the coefficient polynomial "a_{i}" of this algebra's
 226         general characteristic polynomial.
 227
 228         Having this be a separate cached method lets us compute and
 229         store the trace/determinant (a_{r-1} and a_{0} respectively)
 230         separate from the entire characteristic polynomial.
 231         """
 232         (A_of_x, x, xr, detA) = self._charpoly_matrix_system()
 233         R = A_of_x.base_ring()
 234         if i >= self.rank():
 235             # Guaranteed by theory
 236             return R.zero()
 237
 238         # Danger: the in-place modification is done for performance
 239         # reasons (reconstructing a matrix with huge polynomial
 240         # entries is slow), but I don't know how cached_method works,
 241         # so it's highly possible that we're modifying some global
 242         # list variable by reference, here. In other words, you
 243         # probably shouldn't call this method twice on the same
 244         # algebra, at the same time, in two threads
 245         Ai_orig = A_of_x.column(i)
 246         A_of_x.set_column(i,xr)
 247         numerator = A_of_x.det()
 248         A_of_x.set_column(i,Ai_orig)
 249
 250         # We're relying on the theory here to ensure that each a_i is
 251         # indeed back in R, and the added negative signs are to make
 252         # the whole charpoly expression sum to zero.
 253         return R(-numerator/detA)
 254
 255
 256     @cached_method
 257     def _charpoly_matrix_system(self):
 258         """
 259         Compute the matrix whose entries A_ij are polynomials in
 260         X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
 261         corresponding to `x^r` and the determinent of the matrix A =
 262         [A_ij]. In other words, all of the fixed (cachable) data needed
 263         to compute the coefficients of the characteristic polynomial.
 264         """
 265         r = self.rank()
 266         n = self.dimension()
 267
 268         # Construct a new algebra over a multivariate polynomial ring...
 269         names = ['X' + str(i) for i in range(1,n+1)]
 270         R = PolynomialRing(self.base_ring(), names)
 271         J = FiniteDimensionalEuclideanJordanAlgebra(R,
 272                                                     self._multiplication_table,
 273                                                     rank=r)
 274
 275         idmat = identity_matrix(J.base_ring(), n)
 276
 277         W = self._charpoly_basis_space()
 278         W = W.change_ring(R.fraction_field())
 279
 280         # Starting with the standard coordinates x = (X1,X2,...,Xn)
 281         # and then converting the entries to W-coordinates allows us
 282         # to pass in the standard coordinates to the charpoly and get
 283         # back the right answer. Specifically, with x = (X1,X2,...,Xn),
 284         # we have
 285         #
 286         #   W.coordinates(x^2) eval'd at (standard z-coords)
 287         #     =
 288         #   W-coords of (z^2)
 289         #     =
 290         #   W-coords of (standard coords of x^2 eval'd at std-coords of z)
 291         #
 292         # We want the middle equivalent thing in our matrix, but use
 293         # the first equivalent thing instead so that we can pass in
 294         # standard coordinates.
 295         x = J(vector(R, R.gens()))
 296         l1 = [column_matrix(W.coordinates((x**k).vector())) for k in range(r)]
 297         l2 = [idmat.column(k-1).column() for k in range(r+1, n+1)]
 298         A_of_x = block_matrix(R, 1, n, (l1 + l2))
 299         xr = W.coordinates((x**r).vector())
 300         return (A_of_x, x, xr, A_of_x.det())
 301
 302
 303     @cached_method
 304     def characteristic_polynomial(self):
 305         """
 306
 307         .. WARNING::
 308
 309             This implementation doesn't guarantee that the polynomial
 310             denominator in the coefficients is not identically zero, so
 311             theoretically it could crash. The way that this is handled
 312             in e.g. Faraut and Koranyi is to use a basis that guarantees
 313             the denominator is non-zero. But, doing so requires knowledge
 314             of at least one regular element, and we don't even know how
 315             to do that. The trade-off is that, if we use the standard basis,
 316             the resulting polynomial will accept the "usual" coordinates. In
 317             other words, we don't have to do a change of basis before e.g.
 318             computing the trace or determinant.
 319
 320         EXAMPLES:
 321
 322         The characteristic polynomial in the spin algebra is given in
 323         Alizadeh, Example 11.11::
 324
 325             sage: J = JordanSpinEJA(3)
 326             sage: p = J.characteristic_polynomial(); p
 327             X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
 328             sage: xvec = J.one().vector()
 329             sage: p(*xvec)
 330             t^2 - 2*t + 1
 331
 332         """
 333         r = self.rank()
 334         n = self.dimension()
 335
 336         # The list of coefficient polynomials a_1, a_2, ..., a_n.
 337         a = [ self._charpoly_coeff(i) for i in range(n) ]
 338
 339         # We go to a bit of trouble here to reorder the
 340         # indeterminates, so that it's easier to evaluate the
 341         # characteristic polynomial at x's coordinates and get back
 342         # something in terms of t, which is what we want.
 343         R = a[0].parent()
 344         S = PolynomialRing(self.base_ring(),'t')
 345         t = S.gen(0)
 346         S = PolynomialRing(S, R.variable_names())
 347         t = S(t)
 348
 349         # Note: all entries past the rth should be zero. The
 350         # coefficient of the highest power (x^r) is 1, but it doesn't
 351         # appear in the solution vector which contains coefficients
 352         # for the other powers (to make them sum to x^r).
 353         if (r < n):
 354             a[r] = 1 # corresponds to x^r
 355         else:
 356             # When the rank is equal to the dimension, trying to
 357             # assign a[r] goes out-of-bounds.
 358             a.append(1) # corresponds to x^r
 359
 360         return sum( a[k]*(t**k) for k in range(len(a)) )
 361
 362
 363     def inner_product(self, x, y):
 364         """
 365         The inner product associated with this Euclidean Jordan algebra.
 366
 367         Defaults to the trace inner product, but can be overridden by
 368         subclasses if they are sure that the necessary properties are
 369         satisfied.
 370
 371         EXAMPLES:
 372
 373         The inner product must satisfy its axiom for this algebra to truly
 374         be a Euclidean Jordan Algebra::
 375
 376             sage: set_random_seed()
 377             sage: J = random_eja()
 378             sage: x = J.random_element()
 379             sage: y = J.random_element()
 380             sage: z = J.random_element()
 381             sage: (x*y).inner_product(z) == y.inner_product(x*z)
 382             True
 383
 384         """
 385         if (not x in self) or (not y in self):
 386             raise TypeError("arguments must live in this algebra")
 387         return x.trace_inner_product(y)
 388
 389
 390     def natural_basis(self):
 391         """
 392         Return a more-natural representation of this algebra's basis.
 393
 394         Every finite-dimensional Euclidean Jordan Algebra is a direct
 395         sum of five simple algebras, four of which comprise Hermitian
 396         matrices. This method returns the original "natural" basis
 397         for our underlying vector space. (Typically, the natural basis
 398         is used to construct the multiplication table in the first place.)
 399
 400         Note that this will always return a matrix. The standard basis
 401         in `R^n` will be returned as `n`-by-`1` column matrices.
 402
 403         EXAMPLES::
 404
 405             sage: J = RealSymmetricEJA(2)
 406             sage: J.basis()
 407             Family (e0, e1, e2)
 408             sage: J.natural_basis()
 409             (
 410             [1 0]  [0 1]  [0 0]
 411             [0 0], [1 0], [0 1]
 412             )
 413
 414         ::
 415
 416             sage: J = JordanSpinEJA(2)
 417             sage: J.basis()
 418             Family (e0, e1)
 419             sage: J.natural_basis()
 420             (
 421             [1]  [0]
 422             [0], [1]
 423             )
 424
 425         """
 426         if self._natural_basis is None:
 427             return tuple( b.vector().column() for b in self.basis() )
 428         else:
 429             return self._natural_basis
 430
 431
 432     def rank(self):
 433         """
 434         Return the rank of this EJA.
 435         """
 436         if self._rank is None:
 437             raise ValueError("no rank specified at genesis")
 438         else:
 439             return self._rank
 440
 441
 442     class Element(FiniteDimensionalAlgebraElement):
 443         """
 444         An element of a Euclidean Jordan algebra.
 445         """
 446
 447         def __dir__(self):
 448             """
 449             Oh man, I should not be doing this. This hides the "disabled"
 450             methods ``left_matrix`` and ``matrix`` from introspection;
 451             in particular it removes them from tab-completion.
 452             """
 453             return filter(lambda s: s not in ['left_matrix', 'matrix'],
 454                           dir(self.__class__) )
 455
 456
 457         def __init__(self, A, elt=None):
 458             """
 459             EXAMPLES:
 460
 461             The identity in `S^n` is converted to the identity in the EJA::
 462
 463                 sage: J = RealSymmetricEJA(3)
 464                 sage: I = identity_matrix(QQ,3)
 465                 sage: J(I) == J.one()
 466                 True
 467
 468             This skew-symmetric matrix can't be represented in the EJA::
 469
 470                 sage: J = RealSymmetricEJA(3)
 471                 sage: A = matrix(QQ,3, lambda i,j: i-j)
 472                 sage: J(A)
 473                 Traceback (most recent call last):
 474                 ...
 475                 ArithmeticError: vector is not in free module
 476
 477             """
 478             # Goal: if we're given a matrix, and if it lives in our
 479             # parent algebra's "natural ambient space," convert it
 480             # into an algebra element.
 481             #
 482             # The catch is, we make a recursive call after converting
 483             # the given matrix into a vector that lives in the algebra.
 484             # This we need to try the parent class initializer first,
 485             # to avoid recursing forever if we're given something that
 486             # already fits into the algebra, but also happens to live
 487             # in the parent's "natural ambient space" (this happens with
 488             # vectors in R^n).
 489             try:
 490                 FiniteDimensionalAlgebraElement.__init__(self, A, elt)
 491             except ValueError:
 492                 natural_basis = A.natural_basis()
 493                 if elt in natural_basis[0].matrix_space():
 494                     # Thanks for nothing! Matrix spaces aren't vector
 495                     # spaces in Sage, so we have to figure out its
 496                     # natural-basis coordinates ourselves.
 497                     V = VectorSpace(elt.base_ring(), elt.nrows()**2)
 498                     W = V.span( _mat2vec(s) for s in natural_basis )
 499                     coords =  W.coordinates(_mat2vec(elt))
 500                     FiniteDimensionalAlgebraElement.__init__(self, A, coords)
 501
 502         def __pow__(self, n):
 503             """
 504             Return ``self`` raised to the power ``n``.
 505
 506             Jordan algebras are always power-associative; see for
 507             example Faraut and Koranyi, Proposition II.1.2 (ii).
 508
 509             .. WARNING:
 510
 511                 We have to override this because our superclass uses row vectors
 512                 instead of column vectors! We, on the other hand, assume column
 513                 vectors everywhere.
 514
 515             EXAMPLES::
 516
 517                 sage: set_random_seed()
 518                 sage: x = random_eja().random_element()
 519                 sage: x.operator_matrix()*x.vector() == (x^2).vector()
 520                 True
 521
 522             A few examples of power-associativity::
 523
 524                 sage: set_random_seed()
 525                 sage: x = random_eja().random_element()
 526                 sage: x*(x*x)*(x*x) == x^5
 527                 True
 528                 sage: (x*x)*(x*x*x) == x^5
 529                 True
 530
 531             We also know that powers operator-commute (Koecher, Chapter
 532             III, Corollary 1)::
 533
 534                 sage: set_random_seed()
 535                 sage: x = random_eja().random_element()
 536                 sage: m = ZZ.random_element(0,10)
 537                 sage: n = ZZ.random_element(0,10)
 538                 sage: Lxm = (x^m).operator_matrix()
 539                 sage: Lxn = (x^n).operator_matrix()
 540                 sage: Lxm*Lxn == Lxn*Lxm
 541                 True
 542
 543             """
 544             A = self.parent()
 545             if n == 0:
 546                 return A.one()
 547             elif n == 1:
 548                 return self
 549             else:
 550                 return A( (self.operator_matrix()**(n-1))*self.vector() )
 551
 552
 553         def apply_univariate_polynomial(self, p):
 554             """
 555             Apply the univariate polynomial ``p`` to this element.
 556
 557             A priori, SageMath won't allow us to apply a univariate
 558             polynomial to an element of an EJA, because we don't know
 559             that EJAs are rings (they are usually not associative). Of
 560             course, we know that EJAs are power-associative, so the
 561             operation is ultimately kosher. This function sidesteps
 562             the CAS to get the answer we want and expect.
 563
 564             EXAMPLES::
 565
 566                 sage: R = PolynomialRing(QQ, 't')
 567                 sage: t = R.gen(0)
 568                 sage: p = t^4 - t^3 + 5*t - 2
 569                 sage: J = RealCartesianProductEJA(5)
 570                 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
 571                 True
 572
 573             TESTS:
 574
 575             We should always get back an element of the algebra::
 576
 577                 sage: set_random_seed()
 578                 sage: p = PolynomialRing(QQ, 't').random_element()
 579                 sage: J = random_eja()
 580                 sage: x = J.random_element()
 581                 sage: x.apply_univariate_polynomial(p) in J
 582                 True
 583
 584             """
 585             if len(p.variables()) > 1:
 586                 raise ValueError("not a univariate polynomial")
 587             P = self.parent()
 588             R = P.base_ring()
 589             # Convert the coeficcients to the parent's base ring,
 590             # because a priori they might live in an (unnecessarily)
 591             # larger ring for which P.sum() would fail below.
 592             cs = [ R(c) for c in p.coefficients(sparse=False) ]
 593             return P.sum( cs[k]*(self**k) for k in range(len(cs)) )
 594
 595
 596         def characteristic_polynomial(self):
 597             """
 598             Return the characteristic polynomial of this element.
 599
 600             EXAMPLES:
 601
 602             The rank of `R^3` is three, and the minimal polynomial of
 603             the identity element is `(t-1)` from which it follows that
 604             the characteristic polynomial should be `(t-1)^3`::
 605
 606                 sage: J = RealCartesianProductEJA(3)
 607                 sage: J.one().characteristic_polynomial()
 608                 t^3 - 3*t^2 + 3*t - 1
 609
 610             Likewise, the characteristic of the zero element in the
 611             rank-three algebra `R^{n}` should be `t^{3}`::
 612
 613                 sage: J = RealCartesianProductEJA(3)
 614                 sage: J.zero().characteristic_polynomial()
 615                 t^3
 616
 617             The characteristic polynomial of an element should evaluate
 618             to zero on that element::
 619
 620                 sage: set_random_seed()
 621                 sage: x = RealCartesianProductEJA(3).random_element()
 622                 sage: p = x.characteristic_polynomial()
 623                 sage: x.apply_univariate_polynomial(p)
 624                 0
 625
 626             """
 627             p = self.parent().characteristic_polynomial()
 628             return p(*self.vector())
 629
 630
 631         def inner_product(self, other):
 632             """
 633             Return the parent algebra's inner product of myself and ``other``.
 634
 635             EXAMPLES:
 636
 637             The inner product in the Jordan spin algebra is the usual
 638             inner product on `R^n` (this example only works because the
 639             basis for the Jordan algebra is the standard basis in `R^n`)::
 640
 641                 sage: J = JordanSpinEJA(3)
 642                 sage: x = vector(QQ,[1,2,3])
 643                 sage: y = vector(QQ,[4,5,6])
 644                 sage: x.inner_product(y)
 645                 32
 646                 sage: J(x).inner_product(J(y))
 647                 32
 648
 649             The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
 650             multiplication is the usual matrix multiplication in `S^n`,
 651             so the inner product of the identity matrix with itself
 652             should be the `n`::
 653
 654                 sage: J = RealSymmetricEJA(3)
 655                 sage: J.one().inner_product(J.one())
 656                 3
 657
 658             Likewise, the inner product on `C^n` is `<X,Y> =
 659             Re(trace(X*Y))`, where we must necessarily take the real
 660             part because the product of Hermitian matrices may not be
 661             Hermitian::
 662
 663                 sage: J = ComplexHermitianEJA(3)
 664                 sage: J.one().inner_product(J.one())
 665                 3
 666
 667             Ditto for the quaternions::
 668
 669                 sage: J = QuaternionHermitianEJA(3)
 670                 sage: J.one().inner_product(J.one())
 671                 3
 672
 673             TESTS:
 674
 675             Ensure that we can always compute an inner product, and that
 676             it gives us back a real number::
 677
 678                 sage: set_random_seed()
 679                 sage: J = random_eja()
 680                 sage: x = J.random_element()
 681                 sage: y = J.random_element()
 682                 sage: x.inner_product(y) in RR
 683                 True
 684
 685             """
 686             P = self.parent()
 687             if not other in P:
 688                 raise TypeError("'other' must live in the same algebra")
 689
 690             return P.inner_product(self, other)
 691
 692
 693         def operator_commutes_with(self, other):
 694             """
 695             Return whether or not this element operator-commutes
 696             with ``other``.
 697
 698             EXAMPLES:
 699
 700             The definition of a Jordan algebra says that any element
 701             operator-commutes with its square::
 702
 703                 sage: set_random_seed()
 704                 sage: x = random_eja().random_element()
 705                 sage: x.operator_commutes_with(x^2)
 706                 True
 707
 708             TESTS:
 709
 710             Test Lemma 1 from Chapter III of Koecher::
 711
 712                 sage: set_random_seed()
 713                 sage: J = random_eja()
 714                 sage: u = J.random_element()
 715                 sage: v = J.random_element()
 716                 sage: lhs = u.operator_commutes_with(u*v)
 717                 sage: rhs = v.operator_commutes_with(u^2)
 718                 sage: lhs == rhs
 719                 True
 720
 721             """
 722             if not other in self.parent():
 723                 raise TypeError("'other' must live in the same algebra")
 724
 725             A = self.operator_matrix()
 726             B = other.operator_matrix()
 727             return (A*B == B*A)
 728
 729
 730         def det(self):
 731             """
 732             Return my determinant, the product of my eigenvalues.
 733
 734             EXAMPLES::
 735
 736                 sage: J = JordanSpinEJA(2)
 737                 sage: e0,e1 = J.gens()
 738                 sage: x = sum( J.gens() )
 739                 sage: x.det()
 740                 0
 741
 742             ::
 743
 744                 sage: J = JordanSpinEJA(3)
 745                 sage: e0,e1,e2 = J.gens()
 746                 sage: x = sum( J.gens() )
 747                 sage: x.det()
 748                 -1
 749
 750             TESTS:
 751
 752             An element is invertible if and only if its determinant is
 753             non-zero::
 754
 755                 sage: set_random_seed()
 756                 sage: x = random_eja().random_element()
 757                 sage: x.is_invertible() == (x.det() != 0)
 758                 True
 759
 760             """
 761             P = self.parent()
 762             r = P.rank()
 763             p = P._charpoly_coeff(0)
 764             # The _charpoly_coeff function already adds the factor of
 765             # -1 to ensure that _charpoly_coeff(0) is really what
 766             # appears in front of t^{0} in the charpoly. However,
 767             # we want (-1)^r times THAT for the determinant.
 768             return ((-1)**r)*p(*self.vector())
 769
 770
 771         def inverse(self):
 772             """
 773             Return the Jordan-multiplicative inverse of this element.
 774
 775             ALGORITHM:
 776
 777             We appeal to the quadratic representation as in Koecher's
 778             Theorem 12 in Chapter III, Section 5.
 779
 780             EXAMPLES:
 781
 782             The inverse in the spin factor algebra is given in Alizadeh's
 783             Example 11.11::
 784
 785                 sage: set_random_seed()
 786                 sage: n = ZZ.random_element(1,10)
 787                 sage: J = JordanSpinEJA(n)
 788                 sage: x = J.random_element()
 789                 sage: while x.is_zero():
 790                 ....:     x = J.random_element()
 791                 sage: x_vec = x.vector()
 792                 sage: x0 = x_vec[0]
 793                 sage: x_bar = x_vec[1:]
 794                 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
 795                 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
 796                 sage: x_inverse = coeff*inv_vec
 797                 sage: x.inverse() == J(x_inverse)
 798                 True
 799
 800             TESTS:
 801
 802             The identity element is its own inverse::
 803
 804                 sage: set_random_seed()
 805                 sage: J = random_eja()
 806                 sage: J.one().inverse() == J.one()
 807                 True
 808
 809             If an element has an inverse, it acts like one::
 810
 811                 sage: set_random_seed()
 812                 sage: J = random_eja()
 813                 sage: x = J.random_element()
 814                 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
 815                 True
 816
 817             The inverse of the inverse is what we started with::
 818
 819                 sage: set_random_seed()
 820                 sage: J = random_eja()
 821                 sage: x = J.random_element()
 822                 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
 823                 True
 824
 825             The zero element is never invertible::
 826
 827                 sage: set_random_seed()
 828                 sage: J = random_eja().zero().inverse()
 829                 Traceback (most recent call last):
 830                 ...
 831                 ValueError: element is not invertible
 832
 833             """
 834             if not self.is_invertible():
 835                 raise ValueError("element is not invertible")
 836
 837             P = self.parent()
 838             return P(self.quadratic_representation().inverse()*self.vector())
 839
 840
 841         def is_invertible(self):
 842             """
 843             Return whether or not this element is invertible.
 844
 845             We can't use the superclass method because it relies on
 846             the algebra being associative.
 847
 848             ALGORITHM:
 849
 850             The usual way to do this is to check if the determinant is
 851             zero, but we need the characteristic polynomial for the
 852             determinant. The minimal polynomial is a lot easier to get,
 853             so we use Corollary 2 in Chapter V of Koecher to check
 854             whether or not the paren't algebra's zero element is a root
 855             of this element's minimal polynomial.
 856
 857             TESTS:
 858
 859             The identity element is always invertible::
 860
 861                 sage: set_random_seed()
 862                 sage: J = random_eja()
 863                 sage: J.one().is_invertible()
 864                 True
 865
 866             The zero element is never invertible::
 867
 868                 sage: set_random_seed()
 869                 sage: J = random_eja()
 870                 sage: J.zero().is_invertible()
 871                 False
 872
 873             """
 874             zero = self.parent().zero()
 875             p = self.minimal_polynomial()
 876             return not (p(zero) == zero)
 877
 878
 879         def is_nilpotent(self):
 880             """
 881             Return whether or not some power of this element is zero.
 882
 883             The superclass method won't work unless we're in an
 884             associative algebra, and we aren't. However, we generate
 885             an assocoative subalgebra and we're nilpotent there if and
 886             only if we're nilpotent here (probably).
 887
 888             TESTS:
 889
 890             The identity element is never nilpotent::
 891
 892                 sage: set_random_seed()
 893                 sage: random_eja().one().is_nilpotent()
 894                 False
 895
 896             The additive identity is always nilpotent::
 897
 898                 sage: set_random_seed()
 899                 sage: random_eja().zero().is_nilpotent()
 900                 True
 901
 902             """
 903             # The element we're going to call "is_nilpotent()" on.
 904             # Either myself, interpreted as an element of a finite-
 905             # dimensional algebra, or an element of an associative
 906             # subalgebra.
 907             elt = None
 908
 909             if self.parent().is_associative():
 910                 elt = FiniteDimensionalAlgebraElement(self.parent(), self)
 911             else:
 912                 V = self.span_of_powers()
 913                 assoc_subalg = self.subalgebra_generated_by()
 914                 # Mis-design warning: the basis used for span_of_powers()
 915                 # and subalgebra_generated_by() must be the same, and in
 916                 # the same order!
 917                 elt = assoc_subalg(V.coordinates(self.vector()))
 918
 919             # Recursive call, but should work since elt lives in an
 920             # associative algebra.
 921             return elt.is_nilpotent()
 922
 923
 924         def is_regular(self):
 925             """
 926             Return whether or not this is a regular element.
 927
 928             EXAMPLES:
 929
 930             The identity element always has degree one, but any element
 931             linearly-independent from it is regular::
 932
 933                 sage: J = JordanSpinEJA(5)
 934                 sage: J.one().is_regular()
 935                 False
 936                 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
 937                 sage: for x in J.gens():
 938                 ....:     (J.one() + x).is_regular()
 939                 False
 940                 True
 941                 True
 942                 True
 943                 True
 944
 945             """
 946             return self.degree() == self.parent().rank()
 947
 948
 949         def degree(self):
 950             """
 951             Compute the degree of this element the straightforward way
 952             according to the definition; by appending powers to a list
 953             and figuring out its dimension (that is, whether or not
 954             they're linearly dependent).
 955
 956             EXAMPLES::
 957
 958                 sage: J = JordanSpinEJA(4)
 959                 sage: J.one().degree()
 960                 1
 961                 sage: e0,e1,e2,e3 = J.gens()
 962                 sage: (e0 - e1).degree()
 963                 2
 964
 965             In the spin factor algebra (of rank two), all elements that
 966             aren't multiples of the identity are regular::
 967
 968                 sage: set_random_seed()
 969                 sage: n = ZZ.random_element(1,10)
 970                 sage: J = JordanSpinEJA(n)
 971                 sage: x = J.random_element()
 972                 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
 973                 True
 974
 975             """
 976             return self.span_of_powers().dimension()
 977
 978
 979         def left_matrix(self):
 980             """
 981             Our parent class defines ``left_matrix`` and ``matrix``
 982             methods whose names are misleading. We don't want them.
 983             """
 984             raise NotImplementedError("use operator_matrix() instead")
 985
 986         matrix = left_matrix
 987
 988
 989         def minimal_polynomial(self):
 990             """
 991             Return the minimal polynomial of this element,
 992             as a function of the variable `t`.
 993
 994             ALGORITHM:
 995
 996             We restrict ourselves to the associative subalgebra
 997             generated by this element, and then return the minimal
 998             polynomial of this element's operator matrix (in that
 999             subalgebra). This works by Baes Proposition 2.3.16.
1000
1001             TESTS:
1002
1003             The minimal polynomial of the identity and zero elements are
1004             always the same::
1005
1006                 sage: set_random_seed()
1007                 sage: J = random_eja()
1008                 sage: J.one().minimal_polynomial()
1009                 t - 1
1010                 sage: J.zero().minimal_polynomial()
1011                 t
1012
1013             The degree of an element is (by one definition) the degree
1014             of its minimal polynomial::
1015
1016                 sage: set_random_seed()
1017                 sage: x = random_eja().random_element()
1018                 sage: x.degree() == x.minimal_polynomial().degree()
1019                 True
1020
1021             The minimal polynomial and the characteristic polynomial coincide
1022             and are known (see Alizadeh, Example 11.11) for all elements of
1023             the spin factor algebra that aren't scalar multiples of the
1024             identity::
1025
1026                 sage: set_random_seed()
1027                 sage: n = ZZ.random_element(2,10)
1028                 sage: J = JordanSpinEJA(n)
1029                 sage: y = J.random_element()
1030                 sage: while y == y.coefficient(0)*J.one():
1031                 ....:     y = J.random_element()
1032                 sage: y0 = y.vector()[0]
1033                 sage: y_bar = y.vector()[1:]
1034                 sage: actual = y.minimal_polynomial()
1035                 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
1036                 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
1037                 sage: bool(actual == expected)
1038                 True
1039
1040             The minimal polynomial should always kill its element::
1041
1042                 sage: set_random_seed()
1043                 sage: x = random_eja().random_element()
1044                 sage: p = x.minimal_polynomial()
1045                 sage: x.apply_univariate_polynomial(p)
1046                 0
1047
1048             """
1049             V = self.span_of_powers()
1050             assoc_subalg = self.subalgebra_generated_by()
1051             # Mis-design warning: the basis used for span_of_powers()
1052             # and subalgebra_generated_by() must be the same, and in
1053             # the same order!
1054             elt = assoc_subalg(V.coordinates(self.vector()))
1055
1056             # We get back a symbolic polynomial in 'x' but want a real
1057             # polynomial in 't'.
1058             p_of_x = elt.operator_matrix().minimal_polynomial()
1059             return p_of_x.change_variable_name('t')
1060
1061
1062         def natural_representation(self):
1063             """
1064             Return a more-natural representation of this element.
1065
1066             Every finite-dimensional Euclidean Jordan Algebra is a
1067             direct sum of five simple algebras, four of which comprise
1068             Hermitian matrices. This method returns the original
1069             "natural" representation of this element as a Hermitian
1070             matrix, if it has one. If not, you get the usual representation.
1071
1072             EXAMPLES::
1073
1074                 sage: J = ComplexHermitianEJA(3)
1075                 sage: J.one()
1076                 e0 + e5 + e8
1077                 sage: J.one().natural_representation()
1078                 [1 0 0 0 0 0]
1079                 [0 1 0 0 0 0]
1080                 [0 0 1 0 0 0]
1081                 [0 0 0 1 0 0]
1082                 [0 0 0 0 1 0]
1083                 [0 0 0 0 0 1]
1084
1085             ::
1086
1087                 sage: J = QuaternionHermitianEJA(3)
1088                 sage: J.one()
1089                 e0 + e9 + e14
1090                 sage: J.one().natural_representation()
1091                 [1 0 0 0 0 0 0 0 0 0 0 0]
1092                 [0 1 0 0 0 0 0 0 0 0 0 0]
1093                 [0 0 1 0 0 0 0 0 0 0 0 0]
1094                 [0 0 0 1 0 0 0 0 0 0 0 0]
1095                 [0 0 0 0 1 0 0 0 0 0 0 0]
1096                 [0 0 0 0 0 1 0 0 0 0 0 0]
1097                 [0 0 0 0 0 0 1 0 0 0 0 0]
1098                 [0 0 0 0 0 0 0 1 0 0 0 0]
1099                 [0 0 0 0 0 0 0 0 1 0 0 0]
1100                 [0 0 0 0 0 0 0 0 0 1 0 0]
1101                 [0 0 0 0 0 0 0 0 0 0 1 0]
1102                 [0 0 0 0 0 0 0 0 0 0 0 1]
1103
1104             """
1105             B = self.parent().natural_basis()
1106             W = B[0].matrix_space()
1107             return W.linear_combination(zip(self.vector(), B))
1108
1109
1110         def operator(self):
1111             """
1112             Return the left-multiplication-by-this-element
1113             operator on the ambient algebra.
1114
1115             TESTS::
1116
1117                 sage: set_random_seed()
1118                 sage: J = random_eja()
1119                 sage: x = J.random_element()
1120                 sage: y = J.random_element()
1121                 sage: x.operator()(y) == x*y
1122                 True
1123                 sage: y.operator()(x) == x*y
1124                 True
1125
1126             """
1127             P = self.parent()
1128             return FiniteDimensionalEuclideanJordanAlgebraMorphism(
1129                      Hom(P,P),
1130                      self.operator_matrix() )
1131
1132
1133
1134         def operator_matrix(self):
1135             """
1136             Return the matrix that represents left- (or right-)
1137             multiplication by this element in the parent algebra.
1138
1139             We implement this ourselves to work around the fact that
1140             our parent class represents everything with row vectors.
1141
1142             EXAMPLES:
1143
1144             Test the first polarization identity from my notes, Koecher Chapter
1145             III, or from Baes (2.3)::
1146
1147                 sage: set_random_seed()
1148                 sage: J = random_eja()
1149                 sage: x = J.random_element()
1150                 sage: y = J.random_element()
1151                 sage: Lx = x.operator_matrix()
1152                 sage: Ly = y.operator_matrix()
1153                 sage: Lxx = (x*x).operator_matrix()
1154                 sage: Lxy = (x*y).operator_matrix()
1155                 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
1156                 True
1157
1158             Test the second polarization identity from my notes or from
1159             Baes (2.4)::
1160
1161                 sage: set_random_seed()
1162                 sage: J = random_eja()
1163                 sage: x = J.random_element()
1164                 sage: y = J.random_element()
1165                 sage: z = J.random_element()
1166                 sage: Lx = x.operator_matrix()
1167                 sage: Ly = y.operator_matrix()
1168                 sage: Lz = z.operator_matrix()
1169                 sage: Lzy = (z*y).operator_matrix()
1170                 sage: Lxy = (x*y).operator_matrix()
1171                 sage: Lxz = (x*z).operator_matrix()
1172                 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
1173                 True
1174
1175             Test the third polarization identity from my notes or from
1176             Baes (2.5)::
1177
1178                 sage: set_random_seed()
1179                 sage: J = random_eja()
1180                 sage: u = J.random_element()
1181                 sage: y = J.random_element()
1182                 sage: z = J.random_element()
1183                 sage: Lu = u.operator_matrix()
1184                 sage: Ly = y.operator_matrix()
1185                 sage: Lz = z.operator_matrix()
1186                 sage: Lzy = (z*y).operator_matrix()
1187                 sage: Luy = (u*y).operator_matrix()
1188                 sage: Luz = (u*z).operator_matrix()
1189                 sage: Luyz = (u*(y*z)).operator_matrix()
1190                 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
1191                 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
1192                 sage: bool(lhs == rhs)
1193                 True
1194
1195             """
1196             fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self)
1197             return fda_elt.matrix().transpose()
1198
1199
1200         def quadratic_representation(self, other=None):
1201             """
1202             Return the quadratic representation of this element.
1203
1204             EXAMPLES:
1205
1206             The explicit form in the spin factor algebra is given by
1207             Alizadeh's Example 11.12::
1208
1209                 sage: set_random_seed()
1210                 sage: n = ZZ.random_element(1,10)
1211                 sage: J = JordanSpinEJA(n)
1212                 sage: x = J.random_element()
1213                 sage: x_vec = x.vector()
1214                 sage: x0 = x_vec[0]
1215                 sage: x_bar = x_vec[1:]
1216                 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1217                 sage: B = 2*x0*x_bar.row()
1218                 sage: C = 2*x0*x_bar.column()
1219                 sage: D = identity_matrix(QQ, n-1)
1220                 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1221                 sage: D = D + 2*x_bar.tensor_product(x_bar)
1222                 sage: Q = block_matrix(2,2,[A,B,C,D])
1223                 sage: Q == x.quadratic_representation()
1224                 True
1225
1226             Test all of the properties from Theorem 11.2 in Alizadeh::
1227
1228                 sage: set_random_seed()
1229                 sage: J = random_eja()
1230                 sage: x = J.random_element()
1231                 sage: y = J.random_element()
1232                 sage: Lx = x.operator_matrix()
1233                 sage: Lxx = (x*x).operator_matrix()
1234                 sage: Qx = x.quadratic_representation()
1235                 sage: Qy = y.quadratic_representation()
1236                 sage: Qxy = x.quadratic_representation(y)
1237                 sage: Qex = J.one().quadratic_representation(x)
1238                 sage: n = ZZ.random_element(10)
1239                 sage: Qxn = (x^n).quadratic_representation()
1240
1241             Property 1:
1242
1243                 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1244                 True
1245
1246             Property 2:
1247
1248                 sage: alpha = QQ.random_element()
1249                 sage: (alpha*x).quadratic_representation() == (alpha^2)*Qx
1250                 True
1251
1252             Property 3:
1253
1254                 sage: not x.is_invertible() or (
1255                 ....:     Qx*x.inverse().vector() == x.vector() )
1256                 True
1257
1258                 sage: not x.is_invertible() or (
1259                 ....:   Qx.inverse()
1260                 ....:   ==
1261                 ....:   x.inverse().quadratic_representation() )
1262                 True
1263
1264                 sage: Qxy*(J.one().vector()) == (x*y).vector()
1265                 True
1266
1267             Property 4:
1268
1269                 sage: not x.is_invertible() or (
1270                 ....:   x.quadratic_representation(x.inverse())*Qx
1271                 ....:   == Qx*x.quadratic_representation(x.inverse()) )
1272                 True
1273
1274                 sage: not x.is_invertible() or (
1275                 ....:   x.quadratic_representation(x.inverse())*Qx
1276                 ....:   ==
1277                 ....:   2*x.operator_matrix()*Qex - Qx )
1278                 True
1279
1280                 sage: 2*x.operator_matrix()*Qex - Qx == Lxx
1281                 True
1282
1283             Property 5:
1284
1285                 sage: J(Qy*x.vector()).quadratic_representation() == Qy*Qx*Qy
1286                 True
1287
1288             Property 6:
1289
1290                 sage: Qxn == (Qx)^n
1291                 True
1292
1293             Property 7:
1294
1295                 sage: not x.is_invertible() or (
1296                 ....:   Qx*x.inverse().operator_matrix() == Lx )
1297                 True
1298
1299             Property 8:
1300
1301                 sage: not x.operator_commutes_with(y) or (
1302                 ....:   J(Qx*y.vector())^n == J(Qxn*(y^n).vector()) )
1303                 True
1304
1305             """
1306             if other is None:
1307                 other=self
1308             elif not other in self.parent():
1309                 raise TypeError("'other' must live in the same algebra")
1310
1311             L = self.operator_matrix()
1312             M = other.operator_matrix()
1313             return ( L*M + M*L - (self*other).operator_matrix() )
1314
1315
1316         def span_of_powers(self):
1317             """
1318             Return the vector space spanned by successive powers of
1319             this element.
1320             """
1321             # The dimension of the subalgebra can't be greater than
1322             # the big algebra, so just put everything into a list
1323             # and let span() get rid of the excess.
1324             #
1325             # We do the extra ambient_vector_space() in case we're messing
1326             # with polynomials and the direct parent is a module.
1327             V = self.vector().parent().ambient_vector_space()
1328             return V.span( (self**d).vector() for d in xrange(V.dimension()) )
1329
1330
1331         def subalgebra_generated_by(self):
1332             """
1333             Return the associative subalgebra of the parent EJA generated
1334             by this element.
1335
1336             TESTS::
1337
1338                 sage: set_random_seed()
1339                 sage: x = random_eja().random_element()
1340                 sage: x.subalgebra_generated_by().is_associative()
1341                 True
1342
1343             Squaring in the subalgebra should be the same thing as
1344             squaring in the superalgebra::
1345
1346                 sage: set_random_seed()
1347                 sage: x = random_eja().random_element()
1348                 sage: u = x.subalgebra_generated_by().random_element()
1349                 sage: u.operator_matrix()*u.vector() == (u**2).vector()
1350                 True
1351
1352             """
1353             # First get the subspace spanned by the powers of myself...
1354             V = self.span_of_powers()
1355             F = self.base_ring()
1356
1357             # Now figure out the entries of the right-multiplication
1358             # matrix for the successive basis elements b0, b1,... of
1359             # that subspace.
1360             mats = []
1361             for b_right in V.basis():
1362                 eja_b_right = self.parent()(b_right)
1363                 b_right_rows = []
1364                 # The first row of the right-multiplication matrix by
1365                 # b1 is what we get if we apply that matrix to b1. The
1366                 # second row of the right multiplication matrix by b1
1367                 # is what we get when we apply that matrix to b2...
1368                 #
1369                 # IMPORTANT: this assumes that all vectors are COLUMN
1370                 # vectors, unlike our superclass (which uses row vectors).
1371                 for b_left in V.basis():
1372                     eja_b_left = self.parent()(b_left)
1373                     # Multiply in the original EJA, but then get the
1374                     # coordinates from the subalgebra in terms of its
1375                     # basis.
1376                     this_row = V.coordinates((eja_b_left*eja_b_right).vector())
1377                     b_right_rows.append(this_row)
1378                 b_right_matrix = matrix(F, b_right_rows)
1379                 mats.append(b_right_matrix)
1380
1381             # It's an algebra of polynomials in one element, and EJAs
1382             # are power-associative.
1383             #
1384             # TODO: choose generator names intelligently.
1385             return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f')
1386
1387
1388         def subalgebra_idempotent(self):
1389             """
1390             Find an idempotent in the associative subalgebra I generate
1391             using Proposition 2.3.5 in Baes.
1392
1393             TESTS::
1394
1395                 sage: set_random_seed()
1396                 sage: J = random_eja()
1397                 sage: x = J.random_element()
1398                 sage: while x.is_nilpotent():
1399                 ....:     x = J.random_element()
1400                 sage: c = x.subalgebra_idempotent()
1401                 sage: c^2 == c
1402                 True
1403
1404             """
1405             if self.is_nilpotent():
1406                 raise ValueError("this only works with non-nilpotent elements!")
1407
1408             V = self.span_of_powers()
1409             J = self.subalgebra_generated_by()
1410             # Mis-design warning: the basis used for span_of_powers()
1411             # and subalgebra_generated_by() must be the same, and in
1412             # the same order!
1413             u = J(V.coordinates(self.vector()))
1414
1415             # The image of the matrix of left-u^m-multiplication
1416             # will be minimal for some natural number s...
1417             s = 0
1418             minimal_dim = V.dimension()
1419             for i in xrange(1, V.dimension()):
1420                 this_dim = (u**i).operator_matrix().image().dimension()
1421                 if this_dim < minimal_dim:
1422                     minimal_dim = this_dim
1423                     s = i
1424
1425             # Now minimal_matrix should correspond to the smallest
1426             # non-zero subspace in Baes's (or really, Koecher's)
1427             # proposition.
1428             #
1429             # However, we need to restrict the matrix to work on the
1430             # subspace... or do we? Can't we just solve, knowing that
1431             # A(c) = u^(s+1) should have a solution in the big space,
1432             # too?
1433             #
1434             # Beware, solve_right() means that we're using COLUMN vectors.
1435             # Our FiniteDimensionalAlgebraElement superclass uses rows.
1436             u_next = u**(s+1)
1437             A = u_next.operator_matrix()
1438             c_coordinates = A.solve_right(u_next.vector())
1439
1440             # Now c_coordinates is the idempotent we want, but it's in
1441             # the coordinate system of the subalgebra.
1442             #
1443             # We need the basis for J, but as elements of the parent algebra.
1444             #
1445             basis = [self.parent(v) for v in V.basis()]
1446             return self.parent().linear_combination(zip(c_coordinates, basis))
1447
1448
1449         def trace(self):
1450             """
1451             Return my trace, the sum of my eigenvalues.
1452
1453             EXAMPLES::
1454
1455                 sage: J = JordanSpinEJA(3)
1456                 sage: x = sum(J.gens())
1457                 sage: x.trace()
1458                 2
1459
1460             ::
1461
1462                 sage: J = RealCartesianProductEJA(5)
1463                 sage: J.one().trace()
1464                 5
1465
1466             TESTS:
1467
1468             The trace of an element is a real number::
1469
1470                 sage: set_random_seed()
1471                 sage: J = random_eja()
1472                 sage: J.random_element().trace() in J.base_ring()
1473                 True
1474
1475             """
1476             P = self.parent()
1477             r = P.rank()
1478             p = P._charpoly_coeff(r-1)
1479             # The _charpoly_coeff function already adds the factor of
1480             # -1 to ensure that _charpoly_coeff(r-1) is really what
1481             # appears in front of t^{r-1} in the charpoly. However,
1482             # we want the negative of THAT for the trace.
1483             return -p(*self.vector())
1484
1485
1486         def trace_inner_product(self, other):
1487             """
1488             Return the trace inner product of myself and ``other``.
1489
1490             TESTS:
1491
1492             The trace inner product is commutative::
1493
1494                 sage: set_random_seed()
1495                 sage: J = random_eja()
1496                 sage: x = J.random_element(); y = J.random_element()
1497                 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1498                 True
1499
1500             The trace inner product is bilinear::
1501
1502                 sage: set_random_seed()
1503                 sage: J = random_eja()
1504                 sage: x = J.random_element()
1505                 sage: y = J.random_element()
1506                 sage: z = J.random_element()
1507                 sage: a = QQ.random_element();
1508                 sage: actual = (a*(x+z)).trace_inner_product(y)
1509                 sage: expected = ( a*x.trace_inner_product(y) +
1510                 ....:              a*z.trace_inner_product(y) )
1511                 sage: actual == expected
1512                 True
1513                 sage: actual = x.trace_inner_product(a*(y+z))
1514                 sage: expected = ( a*x.trace_inner_product(y) +
1515                 ....:              a*x.trace_inner_product(z) )
1516                 sage: actual == expected
1517                 True
1518
1519             The trace inner product satisfies the compatibility
1520             condition in the definition of a Euclidean Jordan algebra::
1521
1522                 sage: set_random_seed()
1523                 sage: J = random_eja()
1524                 sage: x = J.random_element()
1525                 sage: y = J.random_element()
1526                 sage: z = J.random_element()
1527                 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1528                 True
1529
1530             """
1531             if not other in self.parent():
1532                 raise TypeError("'other' must live in the same algebra")
1533
1534             return (self*other).trace()
1535
1536
1537 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra):
1538     """
1539     Return the Euclidean Jordan Algebra corresponding to the set
1540     `R^n` under the Hadamard product.
1541
1542     Note: this is nothing more than the Cartesian product of ``n``
1543     copies of the spin algebra. Once Cartesian product algebras
1544     are implemented, this can go.
1545
1546     EXAMPLES:
1547
1548     This multiplication table can be verified by hand::
1549
1550         sage: J = RealCartesianProductEJA(3)
1551         sage: e0,e1,e2 = J.gens()
1552         sage: e0*e0
1553         e0
1554         sage: e0*e1
1555         0
1556         sage: e0*e2
1557         0
1558         sage: e1*e1
1559         e1
1560         sage: e1*e2
1561         0
1562         sage: e2*e2
1563         e2
1564
1565     """
1566     @staticmethod
1567     def __classcall_private__(cls, n, field=QQ):
1568         # The FiniteDimensionalAlgebra constructor takes a list of
1569         # matrices, the ith representing right multiplication by the ith
1570         # basis element in the vector space. So if e_1 = (1,0,0), then
1571         # right (Hadamard) multiplication of x by e_1 picks out the first
1572         # component of x; and likewise for the ith basis element e_i.
1573         Qs = [ matrix(field, n, n, lambda k,j: 1*(k == j == i))
1574                for i in xrange(n) ]
1575
1576         fdeja = super(RealCartesianProductEJA, cls)
1577         return fdeja.__classcall_private__(cls, field, Qs, rank=n)
1578
1579     def inner_product(self, x, y):
1580         return _usual_ip(x,y)
1581
1582
1583 def random_eja():
1584     """
1585     Return a "random" finite-dimensional Euclidean Jordan Algebra.
1586
1587     ALGORITHM:
1588
1589     For now, we choose a random natural number ``n`` (greater than zero)
1590     and then give you back one of the following:
1591
1592       * The cartesian product of the rational numbers ``n`` times; this is
1593         ``QQ^n`` with the Hadamard product.
1594
1595       * The Jordan spin algebra on ``QQ^n``.
1596
1597       * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1598         product.
1599
1600       * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1601         in the space of ``2n``-by-``2n`` real symmetric matrices.
1602
1603       * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1604         in the space of ``4n``-by-``4n`` real symmetric matrices.
1605
1606     Later this might be extended to return Cartesian products of the
1607     EJAs above.
1608
1609     TESTS::
1610
1611         sage: random_eja()
1612         Euclidean Jordan algebra of degree...
1613
1614     """
1615
1616     # The max_n component lets us choose different upper bounds on the
1617     # value "n" that gets passed to the constructor. This is needed
1618     # because e.g. R^{10} is reasonable to test, while the Hermitian
1619     # 10-by-10 quaternion matrices are not.
1620     (constructor, max_n) = choice([(RealCartesianProductEJA, 6),
1621                                    (JordanSpinEJA, 6),
1622                                    (RealSymmetricEJA, 5),
1623                                    (ComplexHermitianEJA, 4),
1624                                    (QuaternionHermitianEJA, 3)])
1625     n = ZZ.random_element(1, max_n)
1626     return constructor(n, field=QQ)
1627
1628
1629
1630 def _real_symmetric_basis(n, field=QQ):
1631     """
1632     Return a basis for the space of real symmetric n-by-n matrices.
1633     """
1634     # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1635     # coordinates.
1636     S = []
1637     for i in xrange(n):
1638         for j in xrange(i+1):
1639             Eij = matrix(field, n, lambda k,l: k==i and l==j)
1640             if i == j:
1641                 Sij = Eij
1642             else:
1643                 # Beware, orthogonal but not normalized!
1644                 Sij = Eij + Eij.transpose()
1645             S.append(Sij)
1646     return tuple(S)
1647
1648
1649 def _complex_hermitian_basis(n, field=QQ):
1650     """
1651     Returns a basis for the space of complex Hermitian n-by-n matrices.
1652
1653     TESTS::
1654
1655         sage: set_random_seed()
1656         sage: n = ZZ.random_element(1,5)
1657         sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1658         True
1659
1660     """
1661     F = QuadraticField(-1, 'I')
1662     I = F.gen()
1663
1664     # This is like the symmetric case, but we need to be careful:
1665     #
1666     #   * We want conjugate-symmetry, not just symmetry.
1667     #   * The diagonal will (as a result) be real.
1668     #
1669     S = []
1670     for i in xrange(n):
1671         for j in xrange(i+1):
1672             Eij = matrix(field, n, lambda k,l: k==i and l==j)
1673             if i == j:
1674                 Sij = _embed_complex_matrix(Eij)
1675                 S.append(Sij)
1676             else:
1677                 # Beware, orthogonal but not normalized! The second one
1678                 # has a minus because it's conjugated.
1679                 Sij_real = _embed_complex_matrix(Eij + Eij.transpose())
1680                 S.append(Sij_real)
1681                 Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose())
1682                 S.append(Sij_imag)
1683     return tuple(S)
1684
1685
1686 def _quaternion_hermitian_basis(n, field=QQ):
1687     """
1688     Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1689
1690     TESTS::
1691
1692         sage: set_random_seed()
1693         sage: n = ZZ.random_element(1,5)
1694         sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1695         True
1696
1697     """
1698     Q = QuaternionAlgebra(QQ,-1,-1)
1699     I,J,K = Q.gens()
1700
1701     # This is like the symmetric case, but we need to be careful:
1702     #
1703     #   * We want conjugate-symmetry, not just symmetry.
1704     #   * The diagonal will (as a result) be real.
1705     #
1706     S = []
1707     for i in xrange(n):
1708         for j in xrange(i+1):
1709             Eij = matrix(Q, n, lambda k,l: k==i and l==j)
1710             if i == j:
1711                 Sij = _embed_quaternion_matrix(Eij)
1712                 S.append(Sij)
1713             else:
1714                 # Beware, orthogonal but not normalized! The second,
1715                 # third, and fourth ones have a minus because they're
1716                 # conjugated.
1717                 Sij_real = _embed_quaternion_matrix(Eij + Eij.transpose())
1718                 S.append(Sij_real)
1719                 Sij_I = _embed_quaternion_matrix(I*Eij - I*Eij.transpose())
1720                 S.append(Sij_I)
1721                 Sij_J = _embed_quaternion_matrix(J*Eij - J*Eij.transpose())
1722                 S.append(Sij_J)
1723                 Sij_K = _embed_quaternion_matrix(K*Eij - K*Eij.transpose())
1724                 S.append(Sij_K)
1725     return tuple(S)
1726
1727
1728 def _mat2vec(m):
1729         return vector(m.base_ring(), m.list())
1730
1731 def _vec2mat(v):
1732         return matrix(v.base_ring(), sqrt(v.degree()), v.list())
1733
1734 def _multiplication_table_from_matrix_basis(basis):
1735     """
1736     At least three of the five simple Euclidean Jordan algebras have the
1737     symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1738     multiplication on the right is matrix multiplication. Given a basis
1739     for the underlying matrix space, this function returns a
1740     multiplication table (obtained by looping through the basis
1741     elements) for an algebra of those matrices. A reordered copy
1742     of the basis is also returned to work around the fact that
1743     the ``span()`` in this function will change the order of the basis
1744     from what we think it is, to... something else.
1745     """
1746     # In S^2, for example, we nominally have four coordinates even
1747     # though the space is of dimension three only. The vector space V
1748     # is supposed to hold the entire long vector, and the subspace W
1749     # of V will be spanned by the vectors that arise from symmetric
1750     # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1751     field = basis[0].base_ring()
1752     dimension = basis[0].nrows()
1753
1754     V = VectorSpace(field, dimension**2)
1755     W = V.span( _mat2vec(s) for s in basis )
1756
1757     # Taking the span above reorders our basis (thanks, jerk!) so we
1758     # need to put our "matrix basis" in the same order as the
1759     # (reordered) vector basis.
1760     S = tuple( _vec2mat(b) for b in W.basis() )
1761
1762     Qs = []
1763     for s in S:
1764         # Brute force the multiplication-by-s matrix by looping
1765         # through all elements of the basis and doing the computation
1766         # to find out what the corresponding row should be. BEWARE:
1767         # these multiplication tables won't be symmetric! It therefore
1768         # becomes REALLY IMPORTANT that the underlying algebra
1769         # constructor uses ROW vectors and not COLUMN vectors. That's
1770         # why we're computing rows here and not columns.
1771         Q_rows = []
1772         for t in S:
1773             this_row = _mat2vec((s*t + t*s)/2)
1774             Q_rows.append(W.coordinates(this_row))
1775         Q = matrix(field, W.dimension(), Q_rows)
1776         Qs.append(Q)
1777
1778     return (Qs, S)
1779
1780
1781 def _embed_complex_matrix(M):
1782     """
1783     Embed the n-by-n complex matrix ``M`` into the space of real
1784     matrices of size 2n-by-2n via the map the sends each entry `z = a +
1785     bi` to the block matrix ``[[a,b],[-b,a]]``.
1786
1787     EXAMPLES::
1788
1789         sage: F = QuadraticField(-1,'i')
1790         sage: x1 = F(4 - 2*i)
1791         sage: x2 = F(1 + 2*i)
1792         sage: x3 = F(-i)
1793         sage: x4 = F(6)
1794         sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1795         sage: _embed_complex_matrix(M)
1796         [ 4 -2| 1  2]
1797         [ 2  4|-2  1]
1798         [-----+-----]
1799         [ 0 -1| 6  0]
1800         [ 1  0| 0  6]
1801
1802     TESTS:
1803
1804     Embedding is a homomorphism (isomorphism, in fact)::
1805
1806         sage: set_random_seed()
1807         sage: n = ZZ.random_element(5)
1808         sage: F = QuadraticField(-1, 'i')
1809         sage: X = random_matrix(F, n)
1810         sage: Y = random_matrix(F, n)
1811         sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1812         sage: expected = _embed_complex_matrix(X*Y)
1813         sage: actual == expected
1814         True
1815
1816     """
1817     n = M.nrows()
1818     if M.ncols() != n:
1819         raise ValueError("the matrix 'M' must be square")
1820     field = M.base_ring()
1821     blocks = []
1822     for z in M.list():
1823         a = z.real()
1824         b = z.imag()
1825         blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
1826
1827     # We can drop the imaginaries here.
1828     return block_matrix(field.base_ring(), n, blocks)
1829
1830
1831 def _unembed_complex_matrix(M):
1832     """
1833     The inverse of _embed_complex_matrix().
1834
1835     EXAMPLES::
1836
1837         sage: A = matrix(QQ,[ [ 1,  2,   3,  4],
1838         ....:                 [-2,  1,  -4,  3],
1839         ....:                 [ 9,  10, 11, 12],
1840         ....:                 [-10, 9, -12, 11] ])
1841         sage: _unembed_complex_matrix(A)
1842         [  2*i + 1   4*i + 3]
1843         [ 10*i + 9 12*i + 11]
1844
1845     TESTS:
1846
1847     Unembedding is the inverse of embedding::
1848
1849         sage: set_random_seed()
1850         sage: F = QuadraticField(-1, 'i')
1851         sage: M = random_matrix(F, 3)
1852         sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1853         True
1854
1855     """
1856     n = ZZ(M.nrows())
1857     if M.ncols() != n:
1858         raise ValueError("the matrix 'M' must be square")
1859     if not n.mod(2).is_zero():
1860         raise ValueError("the matrix 'M' must be a complex embedding")
1861
1862     F = QuadraticField(-1, 'i')
1863     i = F.gen()
1864
1865     # Go top-left to bottom-right (reading order), converting every
1866     # 2-by-2 block we see to a single complex element.
1867     elements = []
1868     for k in xrange(n/2):
1869         for j in xrange(n/2):
1870             submat = M[2*k:2*k+2,2*j:2*j+2]
1871             if submat[0,0] != submat[1,1]:
1872                 raise ValueError('bad on-diagonal submatrix')
1873             if submat[0,1] != -submat[1,0]:
1874                 raise ValueError('bad off-diagonal submatrix')
1875             z = submat[0,0] + submat[0,1]*i
1876             elements.append(z)
1877
1878     return matrix(F, n/2, elements)
1879
1880
1881 def _embed_quaternion_matrix(M):
1882     """
1883     Embed the n-by-n quaternion matrix ``M`` into the space of real
1884     matrices of size 4n-by-4n by first sending each quaternion entry
1885     `z = a + bi + cj + dk` to the block-complex matrix
1886     ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1887     a real matrix.
1888
1889     EXAMPLES::
1890
1891         sage: Q = QuaternionAlgebra(QQ,-1,-1)
1892         sage: i,j,k = Q.gens()
1893         sage: x = 1 + 2*i + 3*j + 4*k
1894         sage: M = matrix(Q, 1, [[x]])
1895         sage: _embed_quaternion_matrix(M)
1896         [ 1  2  3  4]
1897         [-2  1 -4  3]
1898         [-3  4  1 -2]
1899         [-4 -3  2  1]
1900
1901     Embedding is a homomorphism (isomorphism, in fact)::
1902
1903         sage: set_random_seed()
1904         sage: n = ZZ.random_element(5)
1905         sage: Q = QuaternionAlgebra(QQ,-1,-1)
1906         sage: X = random_matrix(Q, n)
1907         sage: Y = random_matrix(Q, n)
1908         sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1909         sage: expected = _embed_quaternion_matrix(X*Y)
1910         sage: actual == expected
1911         True
1912
1913     """
1914     quaternions = M.base_ring()
1915     n = M.nrows()
1916     if M.ncols() != n:
1917         raise ValueError("the matrix 'M' must be square")
1918
1919     F = QuadraticField(-1, 'i')
1920     i = F.gen()
1921
1922     blocks = []
1923     for z in M.list():
1924         t = z.coefficient_tuple()
1925         a = t[0]
1926         b = t[1]
1927         c = t[2]
1928         d = t[3]
1929         cplx_matrix = matrix(F, 2, [[ a + b*i, c + d*i],
1930                                     [-c + d*i, a - b*i]])
1931         blocks.append(_embed_complex_matrix(cplx_matrix))
1932
1933     # We should have real entries by now, so use the realest field
1934     # we've got for the return value.
1935     return block_matrix(quaternions.base_ring(), n, blocks)
1936
1937
1938 def _unembed_quaternion_matrix(M):
1939     """
1940     The inverse of _embed_quaternion_matrix().
1941
1942     EXAMPLES::
1943
1944         sage: M = matrix(QQ, [[ 1,  2,  3,  4],
1945         ....:                 [-2,  1, -4,  3],
1946         ....:                 [-3,  4,  1, -2],
1947         ....:                 [-4, -3,  2,  1]])
1948         sage: _unembed_quaternion_matrix(M)
1949         [1 + 2*i + 3*j + 4*k]
1950
1951     TESTS:
1952
1953     Unembedding is the inverse of embedding::
1954
1955         sage: set_random_seed()
1956         sage: Q = QuaternionAlgebra(QQ, -1, -1)
1957         sage: M = random_matrix(Q, 3)
1958         sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1959         True
1960
1961     """
1962     n = ZZ(M.nrows())
1963     if M.ncols() != n:
1964         raise ValueError("the matrix 'M' must be square")
1965     if not n.mod(4).is_zero():
1966         raise ValueError("the matrix 'M' must be a complex embedding")
1967
1968     Q = QuaternionAlgebra(QQ,-1,-1)
1969     i,j,k = Q.gens()
1970
1971     # Go top-left to bottom-right (reading order), converting every
1972     # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1973     # quaternion block.
1974     elements = []
1975     for l in xrange(n/4):
1976         for m in xrange(n/4):
1977             submat = _unembed_complex_matrix(M[4*l:4*l+4,4*m:4*m+4])
1978             if submat[0,0] != submat[1,1].conjugate():
1979                 raise ValueError('bad on-diagonal submatrix')
1980             if submat[0,1] != -submat[1,0].conjugate():
1981                 raise ValueError('bad off-diagonal submatrix')
1982             z  = submat[0,0].real() + submat[0,0].imag()*i
1983             z += submat[0,1].real()*j + submat[0,1].imag()*k
1984             elements.append(z)
1985
1986     return matrix(Q, n/4, elements)
1987
1988
1989 # The usual inner product on R^n.
1990 def _usual_ip(x,y):
1991     return x.vector().inner_product(y.vector())
1992
1993 # The inner product used for the real symmetric simple EJA.
1994 # We keep it as a separate function because e.g. the complex
1995 # algebra uses the same inner product, except divided by 2.
1996 def _matrix_ip(X,Y):
1997     X_mat = X.natural_representation()
1998     Y_mat = Y.natural_representation()
1999     return (X_mat*Y_mat).trace()
2000
2001
2002 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra):
2003     """
2004     The rank-n simple EJA consisting of real symmetric n-by-n
2005     matrices, the usual symmetric Jordan product, and the trace inner
2006     product. It has dimension `(n^2 + n)/2` over the reals.
2007
2008     EXAMPLES::
2009
2010         sage: J = RealSymmetricEJA(2)
2011         sage: e0, e1, e2 = J.gens()
2012         sage: e0*e0
2013         e0
2014         sage: e1*e1
2015         e0 + e2
2016         sage: e2*e2
2017         e2
2018
2019     TESTS:
2020
2021     The degree of this algebra is `(n^2 + n) / 2`::
2022
2023         sage: set_random_seed()
2024         sage: n = ZZ.random_element(1,5)
2025         sage: J = RealSymmetricEJA(n)
2026         sage: J.degree() == (n^2 + n)/2
2027         True
2028
2029     The Jordan multiplication is what we think it is::
2030
2031         sage: set_random_seed()
2032         sage: n = ZZ.random_element(1,5)
2033         sage: J = RealSymmetricEJA(n)
2034         sage: x = J.random_element()
2035         sage: y = J.random_element()
2036         sage: actual = (x*y).natural_representation()
2037         sage: X = x.natural_representation()
2038         sage: Y = y.natural_representation()
2039         sage: expected = (X*Y + Y*X)/2
2040         sage: actual == expected
2041         True
2042         sage: J(expected) == x*y
2043         True
2044
2045     """
2046     @staticmethod
2047     def __classcall_private__(cls, n, field=QQ):
2048         S = _real_symmetric_basis(n, field=field)
2049         (Qs, T) = _multiplication_table_from_matrix_basis(S)
2050
2051         fdeja = super(RealSymmetricEJA, cls)
2052         return fdeja.__classcall_private__(cls,
2053                                            field,
2054                                            Qs,
2055                                            rank=n,
2056                                            natural_basis=T)
2057
2058     def inner_product(self, x, y):
2059         return _matrix_ip(x,y)
2060
2061
2062 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
2063     """
2064     The rank-n simple EJA consisting of complex Hermitian n-by-n
2065     matrices over the real numbers, the usual symmetric Jordan product,
2066     and the real-part-of-trace inner product. It has dimension `n^2` over
2067     the reals.
2068
2069     TESTS:
2070
2071     The degree of this algebra is `n^2`::
2072
2073         sage: set_random_seed()
2074         sage: n = ZZ.random_element(1,5)
2075         sage: J = ComplexHermitianEJA(n)
2076         sage: J.degree() == n^2
2077         True
2078
2079     The Jordan multiplication is what we think it is::
2080
2081         sage: set_random_seed()
2082         sage: n = ZZ.random_element(1,5)
2083         sage: J = ComplexHermitianEJA(n)
2084         sage: x = J.random_element()
2085         sage: y = J.random_element()
2086         sage: actual = (x*y).natural_representation()
2087         sage: X = x.natural_representation()
2088         sage: Y = y.natural_representation()
2089         sage: expected = (X*Y + Y*X)/2
2090         sage: actual == expected
2091         True
2092         sage: J(expected) == x*y
2093         True
2094
2095     """
2096     @staticmethod
2097     def __classcall_private__(cls, n, field=QQ):
2098         S = _complex_hermitian_basis(n)
2099         (Qs, T) = _multiplication_table_from_matrix_basis(S)
2100
2101         fdeja = super(ComplexHermitianEJA, cls)
2102         return fdeja.__classcall_private__(cls,
2103                                            field,
2104                                            Qs,
2105                                            rank=n,
2106                                            natural_basis=T)
2107
2108     def inner_product(self, x, y):
2109         # Since a+bi on the diagonal is represented as
2110         #
2111         #   a + bi  = [  a  b  ]
2112         #             [ -b  a  ],
2113         #
2114         # we'll double-count the "a" entries if we take the trace of
2115         # the embedding.
2116         return _matrix_ip(x,y)/2
2117
2118
2119 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
2120     """
2121     The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
2122     matrices, the usual symmetric Jordan product, and the
2123     real-part-of-trace inner product. It has dimension `2n^2 - n` over
2124     the reals.
2125
2126     TESTS:
2127
2128     The degree of this algebra is `n^2`::
2129
2130         sage: set_random_seed()
2131         sage: n = ZZ.random_element(1,5)
2132         sage: J = QuaternionHermitianEJA(n)
2133         sage: J.degree() == 2*(n^2) - n
2134         True
2135
2136     The Jordan multiplication is what we think it is::
2137
2138         sage: set_random_seed()
2139         sage: n = ZZ.random_element(1,5)
2140         sage: J = QuaternionHermitianEJA(n)
2141         sage: x = J.random_element()
2142         sage: y = J.random_element()
2143         sage: actual = (x*y).natural_representation()
2144         sage: X = x.natural_representation()
2145         sage: Y = y.natural_representation()
2146         sage: expected = (X*Y + Y*X)/2
2147         sage: actual == expected
2148         True
2149         sage: J(expected) == x*y
2150         True
2151
2152     """
2153     @staticmethod
2154     def __classcall_private__(cls, n, field=QQ):
2155         S = _quaternion_hermitian_basis(n)
2156         (Qs, T) = _multiplication_table_from_matrix_basis(S)
2157
2158         fdeja = super(QuaternionHermitianEJA, cls)
2159         return fdeja.__classcall_private__(cls,
2160                                            field,
2161                                            Qs,
2162                                            rank=n,
2163                                            natural_basis=T)
2164
2165     def inner_product(self, x, y):
2166         # Since a+bi+cj+dk on the diagonal is represented as
2167         #
2168         #   a + bi +cj + dk = [  a  b  c  d]
2169         #                     [ -b  a -d  c]
2170         #                     [ -c  d  a -b]
2171         #                     [ -d -c  b  a],
2172         #
2173         # we'll quadruple-count the "a" entries if we take the trace of
2174         # the embedding.
2175         return _matrix_ip(x,y)/4
2176
2177
2178 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra):
2179     """
2180     The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2181     with the usual inner product and jordan product ``x*y =
2182     (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2183     the reals.
2184
2185     EXAMPLES:
2186
2187     This multiplication table can be verified by hand::
2188
2189         sage: J = JordanSpinEJA(4)
2190         sage: e0,e1,e2,e3 = J.gens()
2191         sage: e0*e0
2192         e0
2193         sage: e0*e1
2194         e1
2195         sage: e0*e2
2196         e2
2197         sage: e0*e3
2198         e3
2199         sage: e1*e2
2200         0
2201         sage: e1*e3
2202         0
2203         sage: e2*e3
2204         0
2205
2206     """
2207     @staticmethod
2208     def __classcall_private__(cls, n, field=QQ):
2209         Qs = []
2210         id_matrix = identity_matrix(field, n)
2211         for i in xrange(n):
2212             ei = id_matrix.column(i)
2213             Qi = zero_matrix(field, n)
2214             Qi.set_row(0, ei)
2215             Qi.set_column(0, ei)
2216             Qi += diagonal_matrix(n, [ei[0]]*n)
2217             # The addition of the diagonal matrix adds an extra ei[0] in the
2218             # upper-left corner of the matrix.
2219             Qi[0,0] = Qi[0,0] * ~field(2)
2220             Qs.append(Qi)
2221
2222         # The rank of the spin algebra is two, unless we're in a
2223         # one-dimensional ambient space (because the rank is bounded by
2224         # the ambient dimension).
2225         fdeja = super(JordanSpinEJA, cls)
2226         return fdeja.__classcall_private__(cls, field, Qs, rank=min(n,2))
2227
2228     def inner_product(self, x, y):
2229         return _usual_ip(x,y)