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
  15 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
  16     @staticmethod
  17     def __classcall_private__(cls,
  18                               field,
  19                               mult_table,
  20                               names='e',
  21                               assume_associative=False,
  22                               category=None,
  23                               rank=None,
  24                               natural_basis=None,
  25                               inner_product=None):
  26         n = len(mult_table)
  27         mult_table = [b.base_extend(field) for b in mult_table]
  28         for b in mult_table:
  29             b.set_immutable()
  30             if not (is_Matrix(b) and b.dimensions() == (n, n)):
  31                 raise ValueError("input is not a multiplication table")
  32         mult_table = tuple(mult_table)
  33
  34         cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis()
  35         cat.or_subcategory(category)
  36         if assume_associative:
  37             cat = cat.Associative()
  38
  39         names = normalize_names(n, names)
  40
  41         fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls)
  42         return fda.__classcall__(cls,
  43                                  field,
  44                                  mult_table,
  45                                  assume_associative=assume_associative,
  46                                  names=names,
  47                                  category=cat,
  48                                  rank=rank,
  49                                  natural_basis=natural_basis,
  50                                  inner_product=inner_product)
  51
  52
  53     def __init__(self,
  54                  field,
  55                  mult_table,
  56                  names='e',
  57                  assume_associative=False,
  58                  category=None,
  59                  rank=None,
  60                  natural_basis=None,
  61                  inner_product=None):
  62         """
  63         EXAMPLES:
  64
  65         By definition, Jordan multiplication commutes::
  66
  67             sage: set_random_seed()
  68             sage: J = random_eja()
  69             sage: x = J.random_element()
  70             sage: y = J.random_element()
  71             sage: x*y == y*x
  72             True
  73
  74         """
  75         self._rank = rank
  76         self._natural_basis = natural_basis
  77         self._inner_product = inner_product
  78         fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
  79         fda.__init__(field,
  80                      mult_table,
  81                      names=names,
  82                      category=category)
  83
  84
  85     def _repr_(self):
  86         """
  87         Return a string representation of ``self``.
  88         """
  89         fmt = "Euclidean Jordan algebra of degree {} over {}"
  90         return fmt.format(self.degree(), self.base_ring())
  91
  92
  93     def inner_product(self, x, y):
  94         """
  95         The inner product associated with this Euclidean Jordan algebra.
  96
  97         Will default to the trace inner product if nothing else.
  98
  99         EXAMPLES:
 100
 101         The inner product must satisfy its axiom for this algebra to truly
 102         be a Euclidean Jordan Algebra::
 103
 104             sage: set_random_seed()
 105             sage: J = random_eja()
 106             sage: x = J.random_element()
 107             sage: y = J.random_element()
 108             sage: z = J.random_element()
 109             sage: (x*y).inner_product(z) == y.inner_product(x*z)
 110             True
 111
 112         """
 113         if (not x in self) or (not y in self):
 114             raise TypeError("arguments must live in this algebra")
 115         if self._inner_product is None:
 116             return x.trace_inner_product(y)
 117         else:
 118             return self._inner_product(x,y)
 119
 120
 121     def natural_basis(self):
 122         """
 123         Return a more-natural representation of this algebra's basis.
 124
 125         Every finite-dimensional Euclidean Jordan Algebra is a direct
 126         sum of five simple algebras, four of which comprise Hermitian
 127         matrices. This method returns the original "natural" basis
 128         for our underlying vector space. (Typically, the natural basis
 129         is used to construct the multiplication table in the first place.)
 130
 131         Note that this will always return a matrix. The standard basis
 132         in `R^n` will be returned as `n`-by-`1` column matrices.
 133
 134         EXAMPLES::
 135
 136             sage: J = RealSymmetricEJA(2)
 137             sage: J.basis()
 138             Family (e0, e1, e2)
 139             sage: J.natural_basis()
 140             (
 141             [1 0]  [0 1]  [0 0]
 142             [0 0], [1 0], [0 1]
 143             )
 144
 145         ::
 146
 147             sage: J = JordanSpinEJA(2)
 148             sage: J.basis()
 149             Family (e0, e1)
 150             sage: J.natural_basis()
 151             (
 152             [1]  [0]
 153             [0], [1]
 154             )
 155
 156         """
 157         if self._natural_basis is None:
 158             return tuple( b.vector().column() for b in self.basis() )
 159         else:
 160             return self._natural_basis
 161
 162
 163     def rank(self):
 164         """
 165         Return the rank of this EJA.
 166         """
 167         if self._rank is None:
 168             raise ValueError("no rank specified at genesis")
 169         else:
 170             return self._rank
 171
 172
 173     class Element(FiniteDimensionalAlgebraElement):
 174         """
 175         An element of a Euclidean Jordan algebra.
 176         """
 177
 178         def __init__(self, A, elt=None):
 179             """
 180             EXAMPLES:
 181
 182             The identity in `S^n` is converted to the identity in the EJA::
 183
 184                 sage: J = RealSymmetricEJA(3)
 185                 sage: I = identity_matrix(QQ,3)
 186                 sage: J(I) == J.one()
 187                 True
 188
 189             This skew-symmetric matrix can't be represented in the EJA::
 190
 191                 sage: J = RealSymmetricEJA(3)
 192                 sage: A = matrix(QQ,3, lambda i,j: i-j)
 193                 sage: J(A)
 194                 Traceback (most recent call last):
 195                 ...
 196                 ArithmeticError: vector is not in free module
 197
 198             """
 199             # Goal: if we're given a matrix, and if it lives in our
 200             # parent algebra's "natural ambient space," convert it
 201             # into an algebra element.
 202             #
 203             # The catch is, we make a recursive call after converting
 204             # the given matrix into a vector that lives in the algebra.
 205             # This we need to try the parent class initializer first,
 206             # to avoid recursing forever if we're given something that
 207             # already fits into the algebra, but also happens to live
 208             # in the parent's "natural ambient space" (this happens with
 209             # vectors in R^n).
 210             try:
 211                 FiniteDimensionalAlgebraElement.__init__(self, A, elt)
 212             except ValueError:
 213                 natural_basis = A.natural_basis()
 214                 if elt in natural_basis[0].matrix_space():
 215                     # Thanks for nothing! Matrix spaces aren't vector
 216                     # spaces in Sage, so we have to figure out its
 217                     # natural-basis coordinates ourselves.
 218                     V = VectorSpace(elt.base_ring(), elt.nrows()**2)
 219                     W = V.span( _mat2vec(s) for s in natural_basis )
 220                     coords =  W.coordinates(_mat2vec(elt))
 221                     FiniteDimensionalAlgebraElement.__init__(self, A, coords)
 222
 223         def __pow__(self, n):
 224             """
 225             Return ``self`` raised to the power ``n``.
 226
 227             Jordan algebras are always power-associative; see for
 228             example Faraut and Koranyi, Proposition II.1.2 (ii).
 229
 230             .. WARNING:
 231
 232                 We have to override this because our superclass uses row vectors
 233                 instead of column vectors! We, on the other hand, assume column
 234                 vectors everywhere.
 235
 236             EXAMPLES::
 237
 238                 sage: set_random_seed()
 239                 sage: x = random_eja().random_element()
 240                 sage: x.operator_matrix()*x.vector() == (x^2).vector()
 241                 True
 242
 243             A few examples of power-associativity::
 244
 245                 sage: set_random_seed()
 246                 sage: x = random_eja().random_element()
 247                 sage: x*(x*x)*(x*x) == x^5
 248                 True
 249                 sage: (x*x)*(x*x*x) == x^5
 250                 True
 251
 252             We also know that powers operator-commute (Koecher, Chapter
 253             III, Corollary 1)::
 254
 255                 sage: set_random_seed()
 256                 sage: x = random_eja().random_element()
 257                 sage: m = ZZ.random_element(0,10)
 258                 sage: n = ZZ.random_element(0,10)
 259                 sage: Lxm = (x^m).operator_matrix()
 260                 sage: Lxn = (x^n).operator_matrix()
 261                 sage: Lxm*Lxn == Lxn*Lxm
 262                 True
 263
 264             """
 265             A = self.parent()
 266             if n == 0:
 267                 return A.one()
 268             elif n == 1:
 269                 return self
 270             else:
 271                 return A( (self.operator_matrix()**(n-1))*self.vector() )
 272
 273
 274         def characteristic_polynomial(self):
 275             """
 276             Return my characteristic polynomial (if I'm a regular
 277             element).
 278
 279             Eventually this should be implemented in terms of the parent
 280             algebra's characteristic polynomial that works for ALL
 281             elements.
 282             """
 283             if self.is_regular():
 284                 return self.minimal_polynomial()
 285             else:
 286                 raise NotImplementedError('irregular element')
 287
 288
 289         def inner_product(self, other):
 290             """
 291             Return the parent algebra's inner product of myself and ``other``.
 292
 293             EXAMPLES:
 294
 295             The inner product in the Jordan spin algebra is the usual
 296             inner product on `R^n` (this example only works because the
 297             basis for the Jordan algebra is the standard basis in `R^n`)::
 298
 299                 sage: J = JordanSpinEJA(3)
 300                 sage: x = vector(QQ,[1,2,3])
 301                 sage: y = vector(QQ,[4,5,6])
 302                 sage: x.inner_product(y)
 303                 32
 304                 sage: J(x).inner_product(J(y))
 305                 32
 306
 307             The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
 308             multiplication is the usual matrix multiplication in `S^n`,
 309             so the inner product of the identity matrix with itself
 310             should be the `n`::
 311
 312                 sage: J = RealSymmetricEJA(3)
 313                 sage: J.one().inner_product(J.one())
 314                 3
 315
 316             Likewise, the inner product on `C^n` is `<X,Y> =
 317             Re(trace(X*Y))`, where we must necessarily take the real
 318             part because the product of Hermitian matrices may not be
 319             Hermitian::
 320
 321                 sage: J = ComplexHermitianEJA(3)
 322                 sage: J.one().inner_product(J.one())
 323                 3
 324
 325             Ditto for the quaternions::
 326
 327                 sage: J = QuaternionHermitianEJA(3)
 328                 sage: J.one().inner_product(J.one())
 329                 3
 330
 331             TESTS:
 332
 333             Ensure that we can always compute an inner product, and that
 334             it gives us back a real number::
 335
 336                 sage: set_random_seed()
 337                 sage: J = random_eja()
 338                 sage: x = J.random_element()
 339                 sage: y = J.random_element()
 340                 sage: x.inner_product(y) in RR
 341                 True
 342
 343             """
 344             P = self.parent()
 345             if not other in P:
 346                 raise TypeError("'other' must live in the same algebra")
 347
 348             return P.inner_product(self, other)
 349
 350
 351         def operator_commutes_with(self, other):
 352             """
 353             Return whether or not this element operator-commutes
 354             with ``other``.
 355
 356             EXAMPLES:
 357
 358             The definition of a Jordan algebra says that any element
 359             operator-commutes with its square::
 360
 361                 sage: set_random_seed()
 362                 sage: x = random_eja().random_element()
 363                 sage: x.operator_commutes_with(x^2)
 364                 True
 365
 366             TESTS:
 367
 368             Test Lemma 1 from Chapter III of Koecher::
 369
 370                 sage: set_random_seed()
 371                 sage: J = random_eja()
 372                 sage: u = J.random_element()
 373                 sage: v = J.random_element()
 374                 sage: lhs = u.operator_commutes_with(u*v)
 375                 sage: rhs = v.operator_commutes_with(u^2)
 376                 sage: lhs == rhs
 377                 True
 378
 379             """
 380             if not other in self.parent():
 381                 raise TypeError("'other' must live in the same algebra")
 382
 383             A = self.operator_matrix()
 384             B = other.operator_matrix()
 385             return (A*B == B*A)
 386
 387
 388         def det(self):
 389             """
 390             Return my determinant, the product of my eigenvalues.
 391
 392             EXAMPLES::
 393
 394                 sage: J = JordanSpinEJA(2)
 395                 sage: e0,e1 = J.gens()
 396                 sage: x = e0 + e1
 397                 sage: x.det()
 398                 0
 399                 sage: J = JordanSpinEJA(3)
 400                 sage: e0,e1,e2 = J.gens()
 401                 sage: x = e0 + e1 + e2
 402                 sage: x.det()
 403                 -1
 404
 405             """
 406             cs = self.characteristic_polynomial().coefficients(sparse=False)
 407             r = len(cs) - 1
 408             if r >= 0:
 409                 return cs[0] * (-1)**r
 410             else:
 411                 raise ValueError('charpoly had no coefficients')
 412
 413
 414         def inverse(self):
 415             """
 416             Return the Jordan-multiplicative inverse of this element.
 417
 418             We can't use the superclass method because it relies on the
 419             algebra being associative.
 420
 421             EXAMPLES:
 422
 423             The inverse in the spin factor algebra is given in Alizadeh's
 424             Example 11.11::
 425
 426                 sage: set_random_seed()
 427                 sage: n = ZZ.random_element(1,10)
 428                 sage: J = JordanSpinEJA(n)
 429                 sage: x = J.random_element()
 430                 sage: while x.is_zero():
 431                 ....:     x = J.random_element()
 432                 sage: x_vec = x.vector()
 433                 sage: x0 = x_vec[0]
 434                 sage: x_bar = x_vec[1:]
 435                 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
 436                 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
 437                 sage: x_inverse = coeff*inv_vec
 438                 sage: x.inverse() == J(x_inverse)
 439                 True
 440
 441             TESTS:
 442
 443             The identity element is its own inverse::
 444
 445                 sage: set_random_seed()
 446                 sage: J = random_eja()
 447                 sage: J.one().inverse() == J.one()
 448                 True
 449
 450             If an element has an inverse, it acts like one. TODO: this
 451             can be a lot less ugly once ``is_invertible`` doesn't crash
 452             on irregular elements::
 453
 454                 sage: set_random_seed()
 455                 sage: J = random_eja()
 456                 sage: x = J.random_element()
 457                 sage: try:
 458                 ....:     x.inverse()*x == J.one()
 459                 ....: except:
 460                 ....:     True
 461                 True
 462
 463             """
 464             if self.parent().is_associative():
 465                 elt = FiniteDimensionalAlgebraElement(self.parent(), self)
 466                 return elt.inverse()
 467
 468             # TODO: we can do better once the call to is_invertible()
 469             # doesn't crash on irregular elements.
 470             #if not self.is_invertible():
 471             #    raise ValueError('element is not invertible')
 472
 473             # We do this a little different than the usual recursive
 474             # call to a finite-dimensional algebra element, because we
 475             # wind up with an inverse that lives in the subalgebra and
 476             # we need information about the parent to convert it back.
 477             V = self.span_of_powers()
 478             assoc_subalg = self.subalgebra_generated_by()
 479             # Mis-design warning: the basis used for span_of_powers()
 480             # and subalgebra_generated_by() must be the same, and in
 481             # the same order!
 482             elt = assoc_subalg(V.coordinates(self.vector()))
 483
 484             # This will be in the subalgebra's coordinates...
 485             fda_elt = FiniteDimensionalAlgebraElement(assoc_subalg, elt)
 486             subalg_inverse = fda_elt.inverse()
 487
 488             # So we have to convert back...
 489             basis = [ self.parent(v) for v in V.basis() ]
 490             pairs = zip(subalg_inverse.vector(), basis)
 491             return self.parent().linear_combination(pairs)
 492
 493
 494         def is_invertible(self):
 495             """
 496             Return whether or not this element is invertible.
 497
 498             We can't use the superclass method because it relies on
 499             the algebra being associative.
 500             """
 501             return not self.det().is_zero()
 502
 503
 504         def is_nilpotent(self):
 505             """
 506             Return whether or not some power of this element is zero.
 507
 508             The superclass method won't work unless we're in an
 509             associative algebra, and we aren't. However, we generate
 510             an assocoative subalgebra and we're nilpotent there if and
 511             only if we're nilpotent here (probably).
 512
 513             TESTS:
 514
 515             The identity element is never nilpotent::
 516
 517                 sage: set_random_seed()
 518                 sage: random_eja().one().is_nilpotent()
 519                 False
 520
 521             The additive identity is always nilpotent::
 522
 523                 sage: set_random_seed()
 524                 sage: random_eja().zero().is_nilpotent()
 525                 True
 526
 527             """
 528             # The element we're going to call "is_nilpotent()" on.
 529             # Either myself, interpreted as an element of a finite-
 530             # dimensional algebra, or an element of an associative
 531             # subalgebra.
 532             elt = None
 533
 534             if self.parent().is_associative():
 535                 elt = FiniteDimensionalAlgebraElement(self.parent(), self)
 536             else:
 537                 V = self.span_of_powers()
 538                 assoc_subalg = self.subalgebra_generated_by()
 539                 # Mis-design warning: the basis used for span_of_powers()
 540                 # and subalgebra_generated_by() must be the same, and in
 541                 # the same order!
 542                 elt = assoc_subalg(V.coordinates(self.vector()))
 543
 544             # Recursive call, but should work since elt lives in an
 545             # associative algebra.
 546             return elt.is_nilpotent()
 547
 548
 549         def is_regular(self):
 550             """
 551             Return whether or not this is a regular element.
 552
 553             EXAMPLES:
 554
 555             The identity element always has degree one, but any element
 556             linearly-independent from it is regular::
 557
 558                 sage: J = JordanSpinEJA(5)
 559                 sage: J.one().is_regular()
 560                 False
 561                 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
 562                 sage: for x in J.gens():
 563                 ....:     (J.one() + x).is_regular()
 564                 False
 565                 True
 566                 True
 567                 True
 568                 True
 569
 570             """
 571             return self.degree() == self.parent().rank()
 572
 573
 574         def degree(self):
 575             """
 576             Compute the degree of this element the straightforward way
 577             according to the definition; by appending powers to a list
 578             and figuring out its dimension (that is, whether or not
 579             they're linearly dependent).
 580
 581             EXAMPLES::
 582
 583                 sage: J = JordanSpinEJA(4)
 584                 sage: J.one().degree()
 585                 1
 586                 sage: e0,e1,e2,e3 = J.gens()
 587                 sage: (e0 - e1).degree()
 588                 2
 589
 590             In the spin factor algebra (of rank two), all elements that
 591             aren't multiples of the identity are regular::
 592
 593                 sage: set_random_seed()
 594                 sage: n = ZZ.random_element(1,10)
 595                 sage: J = JordanSpinEJA(n)
 596                 sage: x = J.random_element()
 597                 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
 598                 True
 599
 600             """
 601             return self.span_of_powers().dimension()
 602
 603
 604         def minimal_polynomial(self):
 605             """
 606             EXAMPLES::
 607
 608                 sage: set_random_seed()
 609                 sage: x = random_eja().random_element()
 610                 sage: x.degree() == x.minimal_polynomial().degree()
 611                 True
 612
 613             ::
 614
 615                 sage: set_random_seed()
 616                 sage: x = random_eja().random_element()
 617                 sage: x.degree() == x.minimal_polynomial().degree()
 618                 True
 619
 620             The minimal polynomial and the characteristic polynomial coincide
 621             and are known (see Alizadeh, Example 11.11) for all elements of
 622             the spin factor algebra that aren't scalar multiples of the
 623             identity::
 624
 625                 sage: set_random_seed()
 626                 sage: n = ZZ.random_element(2,10)
 627                 sage: J = JordanSpinEJA(n)
 628                 sage: y = J.random_element()
 629                 sage: while y == y.coefficient(0)*J.one():
 630                 ....:     y = J.random_element()
 631                 sage: y0 = y.vector()[0]
 632                 sage: y_bar = y.vector()[1:]
 633                 sage: actual = y.minimal_polynomial()
 634                 sage: x = SR.symbol('x', domain='real')
 635                 sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2)
 636                 sage: bool(actual == expected)
 637                 True
 638
 639             """
 640             # The element we're going to call "minimal_polynomial()" on.
 641             # Either myself, interpreted as an element of a finite-
 642             # dimensional algebra, or an element of an associative
 643             # subalgebra.
 644             elt = None
 645
 646             if self.parent().is_associative():
 647                 elt = FiniteDimensionalAlgebraElement(self.parent(), self)
 648             else:
 649                 V = self.span_of_powers()
 650                 assoc_subalg = self.subalgebra_generated_by()
 651                 # Mis-design warning: the basis used for span_of_powers()
 652                 # and subalgebra_generated_by() must be the same, and in
 653                 # the same order!
 654                 elt = assoc_subalg(V.coordinates(self.vector()))
 655
 656             # Recursive call, but should work since elt lives in an
 657             # associative algebra.
 658             return elt.minimal_polynomial()
 659
 660
 661         def natural_representation(self):
 662             """
 663             Return a more-natural representation of this element.
 664
 665             Every finite-dimensional Euclidean Jordan Algebra is a
 666             direct sum of five simple algebras, four of which comprise
 667             Hermitian matrices. This method returns the original
 668             "natural" representation of this element as a Hermitian
 669             matrix, if it has one. If not, you get the usual representation.
 670
 671             EXAMPLES::
 672
 673                 sage: J = ComplexHermitianEJA(3)
 674                 sage: J.one()
 675                 e0 + e5 + e8
 676                 sage: J.one().natural_representation()
 677                 [1 0 0 0 0 0]
 678                 [0 1 0 0 0 0]
 679                 [0 0 1 0 0 0]
 680                 [0 0 0 1 0 0]
 681                 [0 0 0 0 1 0]
 682                 [0 0 0 0 0 1]
 683
 684             ::
 685
 686                 sage: J = QuaternionHermitianEJA(3)
 687                 sage: J.one()
 688                 e0 + e9 + e14
 689                 sage: J.one().natural_representation()
 690                 [1 0 0 0 0 0 0 0 0 0 0 0]
 691                 [0 1 0 0 0 0 0 0 0 0 0 0]
 692                 [0 0 1 0 0 0 0 0 0 0 0 0]
 693                 [0 0 0 1 0 0 0 0 0 0 0 0]
 694                 [0 0 0 0 1 0 0 0 0 0 0 0]
 695                 [0 0 0 0 0 1 0 0 0 0 0 0]
 696                 [0 0 0 0 0 0 1 0 0 0 0 0]
 697                 [0 0 0 0 0 0 0 1 0 0 0 0]
 698                 [0 0 0 0 0 0 0 0 1 0 0 0]
 699                 [0 0 0 0 0 0 0 0 0 1 0 0]
 700                 [0 0 0 0 0 0 0 0 0 0 1 0]
 701                 [0 0 0 0 0 0 0 0 0 0 0 1]
 702
 703             """
 704             B = self.parent().natural_basis()
 705             W = B[0].matrix_space()
 706             return W.linear_combination(zip(self.vector(), B))
 707
 708
 709         def operator_matrix(self):
 710             """
 711             Return the matrix that represents left- (or right-)
 712             multiplication by this element in the parent algebra.
 713
 714             We have to override this because the superclass method
 715             returns a matrix that acts on row vectors (that is, on
 716             the right).
 717
 718             EXAMPLES:
 719
 720             Test the first polarization identity from my notes, Koecher Chapter
 721             III, or from Baes (2.3)::
 722
 723                 sage: set_random_seed()
 724                 sage: J = random_eja()
 725                 sage: x = J.random_element()
 726                 sage: y = J.random_element()
 727                 sage: Lx = x.operator_matrix()
 728                 sage: Ly = y.operator_matrix()
 729                 sage: Lxx = (x*x).operator_matrix()
 730                 sage: Lxy = (x*y).operator_matrix()
 731                 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
 732                 True
 733
 734             Test the second polarization identity from my notes or from
 735             Baes (2.4)::
 736
 737                 sage: set_random_seed()
 738                 sage: J = random_eja()
 739                 sage: x = J.random_element()
 740                 sage: y = J.random_element()
 741                 sage: z = J.random_element()
 742                 sage: Lx = x.operator_matrix()
 743                 sage: Ly = y.operator_matrix()
 744                 sage: Lz = z.operator_matrix()
 745                 sage: Lzy = (z*y).operator_matrix()
 746                 sage: Lxy = (x*y).operator_matrix()
 747                 sage: Lxz = (x*z).operator_matrix()
 748                 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
 749                 True
 750
 751             Test the third polarization identity from my notes or from
 752             Baes (2.5)::
 753
 754                 sage: set_random_seed()
 755                 sage: J = random_eja()
 756                 sage: u = J.random_element()
 757                 sage: y = J.random_element()
 758                 sage: z = J.random_element()
 759                 sage: Lu = u.operator_matrix()
 760                 sage: Ly = y.operator_matrix()
 761                 sage: Lz = z.operator_matrix()
 762                 sage: Lzy = (z*y).operator_matrix()
 763                 sage: Luy = (u*y).operator_matrix()
 764                 sage: Luz = (u*z).operator_matrix()
 765                 sage: Luyz = (u*(y*z)).operator_matrix()
 766                 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
 767                 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
 768                 sage: bool(lhs == rhs)
 769                 True
 770
 771             """
 772             fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self)
 773             return fda_elt.matrix().transpose()
 774
 775
 776         def quadratic_representation(self, other=None):
 777             """
 778             Return the quadratic representation of this element.
 779
 780             EXAMPLES:
 781
 782             The explicit form in the spin factor algebra is given by
 783             Alizadeh's Example 11.12::
 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: x_vec = x.vector()
 790                 sage: x0 = x_vec[0]
 791                 sage: x_bar = x_vec[1:]
 792                 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
 793                 sage: B = 2*x0*x_bar.row()
 794                 sage: C = 2*x0*x_bar.column()
 795                 sage: D = identity_matrix(QQ, n-1)
 796                 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
 797                 sage: D = D + 2*x_bar.tensor_product(x_bar)
 798                 sage: Q = block_matrix(2,2,[A,B,C,D])
 799                 sage: Q == x.quadratic_representation()
 800                 True
 801
 802             Test all of the properties from Theorem 11.2 in Alizadeh::
 803
 804                 sage: set_random_seed()
 805                 sage: J = random_eja()
 806                 sage: x = J.random_element()
 807                 sage: y = J.random_element()
 808
 809             Property 1:
 810
 811                 sage: actual = x.quadratic_representation(y)
 812                 sage: expected = ( (x+y).quadratic_representation()
 813                 ....:              -x.quadratic_representation()
 814                 ....:              -y.quadratic_representation() ) / 2
 815                 sage: actual == expected
 816                 True
 817
 818             Property 2:
 819
 820                 sage: alpha = QQ.random_element()
 821                 sage: actual = (alpha*x).quadratic_representation()
 822                 sage: expected = (alpha^2)*x.quadratic_representation()
 823                 sage: actual == expected
 824                 True
 825
 826             Property 5:
 827
 828                 sage: Qy = y.quadratic_representation()
 829                 sage: actual = J(Qy*x.vector()).quadratic_representation()
 830                 sage: expected = Qy*x.quadratic_representation()*Qy
 831                 sage: actual == expected
 832                 True
 833
 834             Property 6:
 835
 836                 sage: k = ZZ.random_element(1,10)
 837                 sage: actual = (x^k).quadratic_representation()
 838                 sage: expected = (x.quadratic_representation())^k
 839                 sage: actual == expected
 840                 True
 841
 842             """
 843             if other is None:
 844                 other=self
 845             elif not other in self.parent():
 846                 raise TypeError("'other' must live in the same algebra")
 847
 848             L = self.operator_matrix()
 849             M = other.operator_matrix()
 850             return ( L*M + M*L - (self*other).operator_matrix() )
 851
 852
 853         def span_of_powers(self):
 854             """
 855             Return the vector space spanned by successive powers of
 856             this element.
 857             """
 858             # The dimension of the subalgebra can't be greater than
 859             # the big algebra, so just put everything into a list
 860             # and let span() get rid of the excess.
 861             V = self.vector().parent()
 862             return V.span( (self**d).vector() for d in xrange(V.dimension()) )
 863
 864
 865         def subalgebra_generated_by(self):
 866             """
 867             Return the associative subalgebra of the parent EJA generated
 868             by this element.
 869
 870             TESTS::
 871
 872                 sage: set_random_seed()
 873                 sage: x = random_eja().random_element()
 874                 sage: x.subalgebra_generated_by().is_associative()
 875                 True
 876
 877             Squaring in the subalgebra should be the same thing as
 878             squaring in the superalgebra::
 879
 880                 sage: set_random_seed()
 881                 sage: x = random_eja().random_element()
 882                 sage: u = x.subalgebra_generated_by().random_element()
 883                 sage: u.operator_matrix()*u.vector() == (u**2).vector()
 884                 True
 885
 886             """
 887             # First get the subspace spanned by the powers of myself...
 888             V = self.span_of_powers()
 889             F = self.base_ring()
 890
 891             # Now figure out the entries of the right-multiplication
 892             # matrix for the successive basis elements b0, b1,... of
 893             # that subspace.
 894             mats = []
 895             for b_right in V.basis():
 896                 eja_b_right = self.parent()(b_right)
 897                 b_right_rows = []
 898                 # The first row of the right-multiplication matrix by
 899                 # b1 is what we get if we apply that matrix to b1. The
 900                 # second row of the right multiplication matrix by b1
 901                 # is what we get when we apply that matrix to b2...
 902                 #
 903                 # IMPORTANT: this assumes that all vectors are COLUMN
 904                 # vectors, unlike our superclass (which uses row vectors).
 905                 for b_left in V.basis():
 906                     eja_b_left = self.parent()(b_left)
 907                     # Multiply in the original EJA, but then get the
 908                     # coordinates from the subalgebra in terms of its
 909                     # basis.
 910                     this_row = V.coordinates((eja_b_left*eja_b_right).vector())
 911                     b_right_rows.append(this_row)
 912                 b_right_matrix = matrix(F, b_right_rows)
 913                 mats.append(b_right_matrix)
 914
 915             # It's an algebra of polynomials in one element, and EJAs
 916             # are power-associative.
 917             #
 918             # TODO: choose generator names intelligently.
 919             return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f')
 920
 921
 922         def subalgebra_idempotent(self):
 923             """
 924             Find an idempotent in the associative subalgebra I generate
 925             using Proposition 2.3.5 in Baes.
 926
 927             TESTS::
 928
 929                 sage: set_random_seed()
 930                 sage: J = eja_rn(5)
 931                 sage: c = J.random_element().subalgebra_idempotent()
 932                 sage: c^2 == c
 933                 True
 934                 sage: J = JordanSpinEJA(5)
 935                 sage: c = J.random_element().subalgebra_idempotent()
 936                 sage: c^2 == c
 937                 True
 938
 939             """
 940             if self.is_nilpotent():
 941                 raise ValueError("this only works with non-nilpotent elements!")
 942
 943             V = self.span_of_powers()
 944             J = self.subalgebra_generated_by()
 945             # Mis-design warning: the basis used for span_of_powers()
 946             # and subalgebra_generated_by() must be the same, and in
 947             # the same order!
 948             u = J(V.coordinates(self.vector()))
 949
 950             # The image of the matrix of left-u^m-multiplication
 951             # will be minimal for some natural number s...
 952             s = 0
 953             minimal_dim = V.dimension()
 954             for i in xrange(1, V.dimension()):
 955                 this_dim = (u**i).operator_matrix().image().dimension()
 956                 if this_dim < minimal_dim:
 957                     minimal_dim = this_dim
 958                     s = i
 959
 960             # Now minimal_matrix should correspond to the smallest
 961             # non-zero subspace in Baes's (or really, Koecher's)
 962             # proposition.
 963             #
 964             # However, we need to restrict the matrix to work on the
 965             # subspace... or do we? Can't we just solve, knowing that
 966             # A(c) = u^(s+1) should have a solution in the big space,
 967             # too?
 968             #
 969             # Beware, solve_right() means that we're using COLUMN vectors.
 970             # Our FiniteDimensionalAlgebraElement superclass uses rows.
 971             u_next = u**(s+1)
 972             A = u_next.operator_matrix()
 973             c_coordinates = A.solve_right(u_next.vector())
 974
 975             # Now c_coordinates is the idempotent we want, but it's in
 976             # the coordinate system of the subalgebra.
 977             #
 978             # We need the basis for J, but as elements of the parent algebra.
 979             #
 980             basis = [self.parent(v) for v in V.basis()]
 981             return self.parent().linear_combination(zip(c_coordinates, basis))
 982
 983
 984         def trace(self):
 985             """
 986             Return my trace, the sum of my eigenvalues.
 987
 988             EXAMPLES::
 989
 990                 sage: J = JordanSpinEJA(3)
 991                 sage: e0,e1,e2 = J.gens()
 992                 sage: x = e0 + e1 + e2
 993                 sage: x.trace()
 994                 2
 995
 996             """
 997             cs = self.characteristic_polynomial().coefficients(sparse=False)
 998             if len(cs) >= 2:
 999                 return -1*cs[-2]
1000             else:
1001                 raise ValueError('charpoly had fewer than 2 coefficients')
1002
1003
1004         def trace_inner_product(self, other):
1005             """
1006             Return the trace inner product of myself and ``other``.
1007             """
1008             if not other in self.parent():
1009                 raise TypeError("'other' must live in the same algebra")
1010
1011             return (self*other).trace()
1012
1013
1014 def eja_rn(dimension, field=QQ):
1015     """
1016     Return the Euclidean Jordan Algebra corresponding to the set
1017     `R^n` under the Hadamard product.
1018
1019     EXAMPLES:
1020
1021     This multiplication table can be verified by hand::
1022
1023         sage: J = eja_rn(3)
1024         sage: e0,e1,e2 = J.gens()
1025         sage: e0*e0
1026         e0
1027         sage: e0*e1
1028         0
1029         sage: e0*e2
1030         0
1031         sage: e1*e1
1032         e1
1033         sage: e1*e2
1034         0
1035         sage: e2*e2
1036         e2
1037
1038     """
1039     # The FiniteDimensionalAlgebra constructor takes a list of
1040     # matrices, the ith representing right multiplication by the ith
1041     # basis element in the vector space. So if e_1 = (1,0,0), then
1042     # right (Hadamard) multiplication of x by e_1 picks out the first
1043     # component of x; and likewise for the ith basis element e_i.
1044     Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
1045            for i in xrange(dimension) ]
1046
1047     return FiniteDimensionalEuclideanJordanAlgebra(field,
1048                                                    Qs,
1049                                                    rank=dimension,
1050                                                    inner_product=_usual_ip)
1051
1052
1053
1054 def random_eja():
1055     """
1056     Return a "random" finite-dimensional Euclidean Jordan Algebra.
1057
1058     ALGORITHM:
1059
1060     For now, we choose a random natural number ``n`` (greater than zero)
1061     and then give you back one of the following:
1062
1063       * The cartesian product of the rational numbers ``n`` times; this is
1064         ``QQ^n`` with the Hadamard product.
1065
1066       * The Jordan spin algebra on ``QQ^n``.
1067
1068       * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1069         product.
1070
1071       * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1072         in the space of ``2n``-by-``2n`` real symmetric matrices.
1073
1074       * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1075         in the space of ``4n``-by-``4n`` real symmetric matrices.
1076
1077     Later this might be extended to return Cartesian products of the
1078     EJAs above.
1079
1080     TESTS::
1081
1082         sage: random_eja()
1083         Euclidean Jordan algebra of degree...
1084
1085     """
1086     n = ZZ.random_element(1,5)
1087     constructor = choice([eja_rn,
1088                           JordanSpinEJA,
1089                           RealSymmetricEJA,
1090                           ComplexHermitianEJA,
1091                           QuaternionHermitianEJA])
1092     return constructor(n, field=QQ)
1093
1094
1095
1096 def _real_symmetric_basis(n, field=QQ):
1097     """
1098     Return a basis for the space of real symmetric n-by-n matrices.
1099     """
1100     # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1101     # coordinates.
1102     S = []
1103     for i in xrange(n):
1104         for j in xrange(i+1):
1105             Eij = matrix(field, n, lambda k,l: k==i and l==j)
1106             if i == j:
1107                 Sij = Eij
1108             else:
1109                 # Beware, orthogonal but not normalized!
1110                 Sij = Eij + Eij.transpose()
1111             S.append(Sij)
1112     return tuple(S)
1113
1114
1115 def _complex_hermitian_basis(n, field=QQ):
1116     """
1117     Returns a basis for the space of complex Hermitian n-by-n matrices.
1118
1119     TESTS::
1120
1121         sage: set_random_seed()
1122         sage: n = ZZ.random_element(1,5)
1123         sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1124         True
1125
1126     """
1127     F = QuadraticField(-1, 'I')
1128     I = F.gen()
1129
1130     # This is like the symmetric case, but we need to be careful:
1131     #
1132     #   * We want conjugate-symmetry, not just symmetry.
1133     #   * The diagonal will (as a result) be real.
1134     #
1135     S = []
1136     for i in xrange(n):
1137         for j in xrange(i+1):
1138             Eij = matrix(field, n, lambda k,l: k==i and l==j)
1139             if i == j:
1140                 Sij = _embed_complex_matrix(Eij)
1141                 S.append(Sij)
1142             else:
1143                 # Beware, orthogonal but not normalized! The second one
1144                 # has a minus because it's conjugated.
1145                 Sij_real = _embed_complex_matrix(Eij + Eij.transpose())
1146                 S.append(Sij_real)
1147                 Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose())
1148                 S.append(Sij_imag)
1149     return tuple(S)
1150
1151
1152 def _quaternion_hermitian_basis(n, field=QQ):
1153     """
1154     Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1155
1156     TESTS::
1157
1158         sage: set_random_seed()
1159         sage: n = ZZ.random_element(1,5)
1160         sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1161         True
1162
1163     """
1164     Q = QuaternionAlgebra(QQ,-1,-1)
1165     I,J,K = Q.gens()
1166
1167     # This is like the symmetric case, but we need to be careful:
1168     #
1169     #   * We want conjugate-symmetry, not just symmetry.
1170     #   * The diagonal will (as a result) be real.
1171     #
1172     S = []
1173     for i in xrange(n):
1174         for j in xrange(i+1):
1175             Eij = matrix(Q, n, lambda k,l: k==i and l==j)
1176             if i == j:
1177                 Sij = _embed_quaternion_matrix(Eij)
1178                 S.append(Sij)
1179             else:
1180                 # Beware, orthogonal but not normalized! The second,
1181                 # third, and fourth ones have a minus because they're
1182                 # conjugated.
1183                 Sij_real = _embed_quaternion_matrix(Eij + Eij.transpose())
1184                 S.append(Sij_real)
1185                 Sij_I = _embed_quaternion_matrix(I*Eij - I*Eij.transpose())
1186                 S.append(Sij_I)
1187                 Sij_J = _embed_quaternion_matrix(J*Eij - J*Eij.transpose())
1188                 S.append(Sij_J)
1189                 Sij_K = _embed_quaternion_matrix(K*Eij - K*Eij.transpose())
1190                 S.append(Sij_K)
1191     return tuple(S)
1192
1193
1194 def _mat2vec(m):
1195         return vector(m.base_ring(), m.list())
1196
1197 def _vec2mat(v):
1198         return matrix(v.base_ring(), sqrt(v.degree()), v.list())
1199
1200 def _multiplication_table_from_matrix_basis(basis):
1201     """
1202     At least three of the five simple Euclidean Jordan algebras have the
1203     symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1204     multiplication on the right is matrix multiplication. Given a basis
1205     for the underlying matrix space, this function returns a
1206     multiplication table (obtained by looping through the basis
1207     elements) for an algebra of those matrices. A reordered copy
1208     of the basis is also returned to work around the fact that
1209     the ``span()`` in this function will change the order of the basis
1210     from what we think it is, to... something else.
1211     """
1212     # In S^2, for example, we nominally have four coordinates even
1213     # though the space is of dimension three only. The vector space V
1214     # is supposed to hold the entire long vector, and the subspace W
1215     # of V will be spanned by the vectors that arise from symmetric
1216     # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1217     field = basis[0].base_ring()
1218     dimension = basis[0].nrows()
1219
1220     V = VectorSpace(field, dimension**2)
1221     W = V.span( _mat2vec(s) for s in basis )
1222
1223     # Taking the span above reorders our basis (thanks, jerk!) so we
1224     # need to put our "matrix basis" in the same order as the
1225     # (reordered) vector basis.
1226     S = tuple( _vec2mat(b) for b in W.basis() )
1227
1228     Qs = []
1229     for s in S:
1230         # Brute force the multiplication-by-s matrix by looping
1231         # through all elements of the basis and doing the computation
1232         # to find out what the corresponding row should be. BEWARE:
1233         # these multiplication tables won't be symmetric! It therefore
1234         # becomes REALLY IMPORTANT that the underlying algebra
1235         # constructor uses ROW vectors and not COLUMN vectors. That's
1236         # why we're computing rows here and not columns.
1237         Q_rows = []
1238         for t in S:
1239             this_row = _mat2vec((s*t + t*s)/2)
1240             Q_rows.append(W.coordinates(this_row))
1241         Q = matrix(field, W.dimension(), Q_rows)
1242         Qs.append(Q)
1243
1244     return (Qs, S)
1245
1246
1247 def _embed_complex_matrix(M):
1248     """
1249     Embed the n-by-n complex matrix ``M`` into the space of real
1250     matrices of size 2n-by-2n via the map the sends each entry `z = a +
1251     bi` to the block matrix ``[[a,b],[-b,a]]``.
1252
1253     EXAMPLES::
1254
1255         sage: F = QuadraticField(-1,'i')
1256         sage: x1 = F(4 - 2*i)
1257         sage: x2 = F(1 + 2*i)
1258         sage: x3 = F(-i)
1259         sage: x4 = F(6)
1260         sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1261         sage: _embed_complex_matrix(M)
1262         [ 4 -2| 1  2]
1263         [ 2  4|-2  1]
1264         [-----+-----]
1265         [ 0 -1| 6  0]
1266         [ 1  0| 0  6]
1267
1268     TESTS:
1269
1270     Embedding is a homomorphism (isomorphism, in fact)::
1271
1272         sage: set_random_seed()
1273         sage: n = ZZ.random_element(5)
1274         sage: F = QuadraticField(-1, 'i')
1275         sage: X = random_matrix(F, n)
1276         sage: Y = random_matrix(F, n)
1277         sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1278         sage: expected = _embed_complex_matrix(X*Y)
1279         sage: actual == expected
1280         True
1281
1282     """
1283     n = M.nrows()
1284     if M.ncols() != n:
1285         raise ValueError("the matrix 'M' must be square")
1286     field = M.base_ring()
1287     blocks = []
1288     for z in M.list():
1289         a = z.real()
1290         b = z.imag()
1291         blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
1292
1293     # We can drop the imaginaries here.
1294     return block_matrix(field.base_ring(), n, blocks)
1295
1296
1297 def _unembed_complex_matrix(M):
1298     """
1299     The inverse of _embed_complex_matrix().
1300
1301     EXAMPLES::
1302
1303         sage: A = matrix(QQ,[ [ 1,  2,   3,  4],
1304         ....:                 [-2,  1,  -4,  3],
1305         ....:                 [ 9,  10, 11, 12],
1306         ....:                 [-10, 9, -12, 11] ])
1307         sage: _unembed_complex_matrix(A)
1308         [  2*i + 1   4*i + 3]
1309         [ 10*i + 9 12*i + 11]
1310
1311     TESTS:
1312
1313     Unembedding is the inverse of embedding::
1314
1315         sage: set_random_seed()
1316         sage: F = QuadraticField(-1, 'i')
1317         sage: M = random_matrix(F, 3)
1318         sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1319         True
1320
1321     """
1322     n = ZZ(M.nrows())
1323     if M.ncols() != n:
1324         raise ValueError("the matrix 'M' must be square")
1325     if not n.mod(2).is_zero():
1326         raise ValueError("the matrix 'M' must be a complex embedding")
1327
1328     F = QuadraticField(-1, 'i')
1329     i = F.gen()
1330
1331     # Go top-left to bottom-right (reading order), converting every
1332     # 2-by-2 block we see to a single complex element.
1333     elements = []
1334     for k in xrange(n/2):
1335         for j in xrange(n/2):
1336             submat = M[2*k:2*k+2,2*j:2*j+2]
1337             if submat[0,0] != submat[1,1]:
1338                 raise ValueError('bad on-diagonal submatrix')
1339             if submat[0,1] != -submat[1,0]:
1340                 raise ValueError('bad off-diagonal submatrix')
1341             z = submat[0,0] + submat[0,1]*i
1342             elements.append(z)
1343
1344     return matrix(F, n/2, elements)
1345
1346
1347 def _embed_quaternion_matrix(M):
1348     """
1349     Embed the n-by-n quaternion matrix ``M`` into the space of real
1350     matrices of size 4n-by-4n by first sending each quaternion entry
1351     `z = a + bi + cj + dk` to the block-complex matrix
1352     ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1353     a real matrix.
1354
1355     EXAMPLES::
1356
1357         sage: Q = QuaternionAlgebra(QQ,-1,-1)
1358         sage: i,j,k = Q.gens()
1359         sage: x = 1 + 2*i + 3*j + 4*k
1360         sage: M = matrix(Q, 1, [[x]])
1361         sage: _embed_quaternion_matrix(M)
1362         [ 1  2  3  4]
1363         [-2  1 -4  3]
1364         [-3  4  1 -2]
1365         [-4 -3  2  1]
1366
1367     Embedding is a homomorphism (isomorphism, in fact)::
1368
1369         sage: set_random_seed()
1370         sage: n = ZZ.random_element(5)
1371         sage: Q = QuaternionAlgebra(QQ,-1,-1)
1372         sage: X = random_matrix(Q, n)
1373         sage: Y = random_matrix(Q, n)
1374         sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1375         sage: expected = _embed_quaternion_matrix(X*Y)
1376         sage: actual == expected
1377         True
1378
1379     """
1380     quaternions = M.base_ring()
1381     n = M.nrows()
1382     if M.ncols() != n:
1383         raise ValueError("the matrix 'M' must be square")
1384
1385     F = QuadraticField(-1, 'i')
1386     i = F.gen()
1387
1388     blocks = []
1389     for z in M.list():
1390         t = z.coefficient_tuple()
1391         a = t[0]
1392         b = t[1]
1393         c = t[2]
1394         d = t[3]
1395         cplx_matrix = matrix(F, 2, [[ a + b*i, c + d*i],
1396                                     [-c + d*i, a - b*i]])
1397         blocks.append(_embed_complex_matrix(cplx_matrix))
1398
1399     # We should have real entries by now, so use the realest field
1400     # we've got for the return value.
1401     return block_matrix(quaternions.base_ring(), n, blocks)
1402
1403
1404 def _unembed_quaternion_matrix(M):
1405     """
1406     The inverse of _embed_quaternion_matrix().
1407
1408     EXAMPLES::
1409
1410         sage: M = matrix(QQ, [[ 1,  2,  3,  4],
1411         ....:                 [-2,  1, -4,  3],
1412         ....:                 [-3,  4,  1, -2],
1413         ....:                 [-4, -3,  2,  1]])
1414         sage: _unembed_quaternion_matrix(M)
1415         [1 + 2*i + 3*j + 4*k]
1416
1417     TESTS:
1418
1419     Unembedding is the inverse of embedding::
1420
1421         sage: set_random_seed()
1422         sage: Q = QuaternionAlgebra(QQ, -1, -1)
1423         sage: M = random_matrix(Q, 3)
1424         sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1425         True
1426
1427     """
1428     n = ZZ(M.nrows())
1429     if M.ncols() != n:
1430         raise ValueError("the matrix 'M' must be square")
1431     if not n.mod(4).is_zero():
1432         raise ValueError("the matrix 'M' must be a complex embedding")
1433
1434     Q = QuaternionAlgebra(QQ,-1,-1)
1435     i,j,k = Q.gens()
1436
1437     # Go top-left to bottom-right (reading order), converting every
1438     # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1439     # quaternion block.
1440     elements = []
1441     for l in xrange(n/4):
1442         for m in xrange(n/4):
1443             submat = _unembed_complex_matrix(M[4*l:4*l+4,4*m:4*m+4])
1444             if submat[0,0] != submat[1,1].conjugate():
1445                 raise ValueError('bad on-diagonal submatrix')
1446             if submat[0,1] != -submat[1,0].conjugate():
1447                 raise ValueError('bad off-diagonal submatrix')
1448             z  = submat[0,0].real() + submat[0,0].imag()*i
1449             z += submat[0,1].real()*j + submat[0,1].imag()*k
1450             elements.append(z)
1451
1452     return matrix(Q, n/4, elements)
1453
1454
1455 # The usual inner product on R^n.
1456 def _usual_ip(x,y):
1457     return x.vector().inner_product(y.vector())
1458
1459 # The inner product used for the real symmetric simple EJA.
1460 # We keep it as a separate function because e.g. the complex
1461 # algebra uses the same inner product, except divided by 2.
1462 def _matrix_ip(X,Y):
1463     X_mat = X.natural_representation()
1464     Y_mat = Y.natural_representation()
1465     return (X_mat*Y_mat).trace()
1466
1467
1468 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra):
1469     """
1470     The rank-n simple EJA consisting of real symmetric n-by-n
1471     matrices, the usual symmetric Jordan product, and the trace inner
1472     product. It has dimension `(n^2 + n)/2` over the reals.
1473
1474     EXAMPLES::
1475
1476         sage: J = RealSymmetricEJA(2)
1477         sage: e0, e1, e2 = J.gens()
1478         sage: e0*e0
1479         e0
1480         sage: e1*e1
1481         e0 + e2
1482         sage: e2*e2
1483         e2
1484
1485     TESTS:
1486
1487     The degree of this algebra is `(n^2 + n) / 2`::
1488
1489         sage: set_random_seed()
1490         sage: n = ZZ.random_element(1,5)
1491         sage: J = RealSymmetricEJA(n)
1492         sage: J.degree() == (n^2 + n)/2
1493         True
1494
1495     The Jordan multiplication is what we think it is::
1496
1497         sage: set_random_seed()
1498         sage: n = ZZ.random_element(1,5)
1499         sage: J = RealSymmetricEJA(n)
1500         sage: x = J.random_element()
1501         sage: y = J.random_element()
1502         sage: actual = (x*y).natural_representation()
1503         sage: X = x.natural_representation()
1504         sage: Y = y.natural_representation()
1505         sage: expected = (X*Y + Y*X)/2
1506         sage: actual == expected
1507         True
1508         sage: J(expected) == x*y
1509         True
1510
1511     """
1512     @staticmethod
1513     def __classcall_private__(cls, n, field=QQ):
1514         S = _real_symmetric_basis(n, field=field)
1515         (Qs, T) = _multiplication_table_from_matrix_basis(S)
1516
1517         fdeja = super(RealSymmetricEJA, cls)
1518         return fdeja.__classcall_private__(cls,
1519                                            field,
1520                                            Qs,
1521                                            rank=n,
1522                                            natural_basis=T)
1523
1524     def inner_product(self, x, y):
1525         return _matrix_ip(x,y)
1526
1527
1528 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
1529     """
1530     The rank-n simple EJA consisting of complex Hermitian n-by-n
1531     matrices over the real numbers, the usual symmetric Jordan product,
1532     and the real-part-of-trace inner product. It has dimension `n^2` over
1533     the reals.
1534
1535     TESTS:
1536
1537     The degree of this algebra is `n^2`::
1538
1539         sage: set_random_seed()
1540         sage: n = ZZ.random_element(1,5)
1541         sage: J = ComplexHermitianEJA(n)
1542         sage: J.degree() == n^2
1543         True
1544
1545     The Jordan multiplication is what we think it is::
1546
1547         sage: set_random_seed()
1548         sage: n = ZZ.random_element(1,5)
1549         sage: J = ComplexHermitianEJA(n)
1550         sage: x = J.random_element()
1551         sage: y = J.random_element()
1552         sage: actual = (x*y).natural_representation()
1553         sage: X = x.natural_representation()
1554         sage: Y = y.natural_representation()
1555         sage: expected = (X*Y + Y*X)/2
1556         sage: actual == expected
1557         True
1558         sage: J(expected) == x*y
1559         True
1560
1561     """
1562     @staticmethod
1563     def __classcall_private__(cls, n, field=QQ):
1564         S = _complex_hermitian_basis(n)
1565         (Qs, T) = _multiplication_table_from_matrix_basis(S)
1566
1567         fdeja = super(ComplexHermitianEJA, cls)
1568         return fdeja.__classcall_private__(cls,
1569                                            field,
1570                                            Qs,
1571                                            rank=n,
1572                                            natural_basis=T)
1573
1574     def inner_product(self, x, y):
1575         # Since a+bi on the diagonal is represented as
1576         #
1577         #   a + bi  = [  a  b  ]
1578         #             [ -b  a  ],
1579         #
1580         # we'll double-count the "a" entries if we take the trace of
1581         # the embedding.
1582         return _matrix_ip(x,y)/2
1583
1584
1585 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
1586     """
1587     The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1588     matrices, the usual symmetric Jordan product, and the
1589     real-part-of-trace inner product. It has dimension `2n^2 - n` over
1590     the reals.
1591
1592     TESTS:
1593
1594     The degree of this algebra is `n^2`::
1595
1596         sage: set_random_seed()
1597         sage: n = ZZ.random_element(1,5)
1598         sage: J = QuaternionHermitianEJA(n)
1599         sage: J.degree() == 2*(n^2) - n
1600         True
1601
1602     The Jordan multiplication is what we think it is::
1603
1604         sage: set_random_seed()
1605         sage: n = ZZ.random_element(1,5)
1606         sage: J = QuaternionHermitianEJA(n)
1607         sage: x = J.random_element()
1608         sage: y = J.random_element()
1609         sage: actual = (x*y).natural_representation()
1610         sage: X = x.natural_representation()
1611         sage: Y = y.natural_representation()
1612         sage: expected = (X*Y + Y*X)/2
1613         sage: actual == expected
1614         True
1615         sage: J(expected) == x*y
1616         True
1617
1618     """
1619     @staticmethod
1620     def __classcall_private__(cls, n, field=QQ):
1621         S = _quaternion_hermitian_basis(n)
1622         (Qs, T) = _multiplication_table_from_matrix_basis(S)
1623
1624         fdeja = super(QuaternionHermitianEJA, cls)
1625         return fdeja.__classcall_private__(cls,
1626                                            field,
1627                                            Qs,
1628                                            rank=n,
1629                                            natural_basis=T)
1630
1631     def inner_product(self, x, y):
1632         # Since a+bi+cj+dk on the diagonal is represented as
1633         #
1634         #   a + bi +cj + dk = [  a  b  c  d]
1635         #                     [ -b  a -d  c]
1636         #                     [ -c  d  a -b]
1637         #                     [ -d -c  b  a],
1638         #
1639         # we'll quadruple-count the "a" entries if we take the trace of
1640         # the embedding.
1641         return _matrix_ip(x,y)/4
1642
1643
1644 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra):
1645     """
1646     The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1647     with the usual inner product and jordan product ``x*y =
1648     (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1649     the reals.
1650
1651     EXAMPLES:
1652
1653     This multiplication table can be verified by hand::
1654
1655         sage: J = JordanSpinEJA(4)
1656         sage: e0,e1,e2,e3 = J.gens()
1657         sage: e0*e0
1658         e0
1659         sage: e0*e1
1660         e1
1661         sage: e0*e2
1662         e2
1663         sage: e0*e3
1664         e3
1665         sage: e1*e2
1666         0
1667         sage: e1*e3
1668         0
1669         sage: e2*e3
1670         0
1671
1672     """
1673     @staticmethod
1674     def __classcall_private__(cls, n, field=QQ):
1675         Qs = []
1676         id_matrix = identity_matrix(field, n)
1677         for i in xrange(n):
1678             ei = id_matrix.column(i)
1679             Qi = zero_matrix(field, n)
1680             Qi.set_row(0, ei)
1681             Qi.set_column(0, ei)
1682             Qi += diagonal_matrix(n, [ei[0]]*n)
1683             # The addition of the diagonal matrix adds an extra ei[0] in the
1684             # upper-left corner of the matrix.
1685             Qi[0,0] = Qi[0,0] * ~field(2)
1686             Qs.append(Qi)
1687
1688         fdeja = super(JordanSpinEJA, cls)
1689         return fdeja.__classcall_private__(cls, field, Qs)
1690
1691     def rank(self):
1692         """
1693         Return the rank of this Jordan Spin Algebra.
1694
1695         The rank of the spin algebra is two, unless we're in a
1696         one-dimensional ambient space (because the rank is bounded by
1697         the ambient dimension).
1698         """
1699         return min(self.dimension(),2)
1700
1701     def inner_product(self, x, y):
1702         return _usual_ip(x,y)