mjo/eja/eja_element.py

   1 from sage.matrix.constructor import matrix
   2 from sage.modules.free_module import VectorSpace
   3 from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
   4
   5 # TODO: make this unnecessary somehow.
   6 from sage.misc.lazy_import import lazy_import
   7 lazy_import('mjo.eja.eja_algebra', 'FiniteDimensionalEuclideanJordanAlgebra')
   8 lazy_import('mjo.eja.eja_subalgebra',
   9             'FiniteDimensionalEuclideanJordanElementSubalgebra')
  10 from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
  11 from mjo.eja.eja_utils import _mat2vec
  12
  13 class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
  14     """
  15     An element of a Euclidean Jordan algebra.
  16     """
  17
  18     def __dir__(self):
  19         """
  20         Oh man, I should not be doing this. This hides the "disabled"
  21         methods ``left_matrix`` and ``matrix`` from introspection;
  22         in particular it removes them from tab-completion.
  23         """
  24         return filter(lambda s: s not in ['left_matrix', 'matrix'],
  25                       dir(self.__class__) )
  26
  27
  28
  29
  30     def __pow__(self, n):
  31         """
  32         Return ``self`` raised to the power ``n``.
  33
  34         Jordan algebras are always power-associative; see for
  35         example Faraut and Koranyi, Proposition II.1.2 (ii).
  36
  37         We have to override this because our superclass uses row
  38         vectors instead of column vectors! We, on the other hand,
  39         assume column vectors everywhere.
  40
  41         SETUP::
  42
  43             sage: from mjo.eja.eja_algebra import random_eja
  44
  45         TESTS:
  46
  47         The definition of `x^2` is the unambiguous `x*x`::
  48
  49             sage: set_random_seed()
  50             sage: x = random_eja().random_element()
  51             sage: x*x == (x^2)
  52             True
  53
  54         A few examples of power-associativity::
  55
  56             sage: set_random_seed()
  57             sage: x = random_eja().random_element()
  58             sage: x*(x*x)*(x*x) == x^5
  59             True
  60             sage: (x*x)*(x*x*x) == x^5
  61             True
  62
  63         We also know that powers operator-commute (Koecher, Chapter
  64         III, Corollary 1)::
  65
  66             sage: set_random_seed()
  67             sage: x = random_eja().random_element()
  68             sage: m = ZZ.random_element(0,10)
  69             sage: n = ZZ.random_element(0,10)
  70             sage: Lxm = (x^m).operator()
  71             sage: Lxn = (x^n).operator()
  72             sage: Lxm*Lxn == Lxn*Lxm
  73             True
  74
  75         """
  76         if n == 0:
  77             return self.parent().one()
  78         elif n == 1:
  79             return self
  80         else:
  81             return (self.operator()**(n-1))(self)
  82
  83
  84     def apply_univariate_polynomial(self, p):
  85         """
  86         Apply the univariate polynomial ``p`` to this element.
  87
  88         A priori, SageMath won't allow us to apply a univariate
  89         polynomial to an element of an EJA, because we don't know
  90         that EJAs are rings (they are usually not associative). Of
  91         course, we know that EJAs are power-associative, so the
  92         operation is ultimately kosher. This function sidesteps
  93         the CAS to get the answer we want and expect.
  94
  95         SETUP::
  96
  97             sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
  98             ....:                                  random_eja)
  99
 100         EXAMPLES::
 101
 102             sage: R = PolynomialRing(QQ, 't')
 103             sage: t = R.gen(0)
 104             sage: p = t^4 - t^3 + 5*t - 2
 105             sage: J = RealCartesianProductEJA(5)
 106             sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
 107             True
 108
 109         TESTS:
 110
 111         We should always get back an element of the algebra::
 112
 113             sage: set_random_seed()
 114             sage: p = PolynomialRing(QQ, 't').random_element()
 115             sage: J = random_eja()
 116             sage: x = J.random_element()
 117             sage: x.apply_univariate_polynomial(p) in J
 118             True
 119
 120         """
 121         if len(p.variables()) > 1:
 122             raise ValueError("not a univariate polynomial")
 123         P = self.parent()
 124         R = P.base_ring()
 125         # Convert the coeficcients to the parent's base ring,
 126         # because a priori they might live in an (unnecessarily)
 127         # larger ring for which P.sum() would fail below.
 128         cs = [ R(c) for c in p.coefficients(sparse=False) ]
 129         return P.sum( cs[k]*(self**k) for k in range(len(cs)) )
 130
 131
 132     def characteristic_polynomial(self):
 133         """
 134         Return the characteristic polynomial of this element.
 135
 136         SETUP::
 137
 138             sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
 139
 140         EXAMPLES:
 141
 142         The rank of `R^3` is three, and the minimal polynomial of
 143         the identity element is `(t-1)` from which it follows that
 144         the characteristic polynomial should be `(t-1)^3`::
 145
 146             sage: J = RealCartesianProductEJA(3)
 147             sage: J.one().characteristic_polynomial()
 148             t^3 - 3*t^2 + 3*t - 1
 149
 150         Likewise, the characteristic of the zero element in the
 151         rank-three algebra `R^{n}` should be `t^{3}`::
 152
 153             sage: J = RealCartesianProductEJA(3)
 154             sage: J.zero().characteristic_polynomial()
 155             t^3
 156
 157         TESTS:
 158
 159         The characteristic polynomial of an element should evaluate
 160         to zero on that element::
 161
 162             sage: set_random_seed()
 163             sage: x = RealCartesianProductEJA(3).random_element()
 164             sage: p = x.characteristic_polynomial()
 165             sage: x.apply_univariate_polynomial(p)
 166             0
 167
 168         The characteristic polynomials of the zero and unit elements
 169         should be what we think they are in a subalgebra, too::
 170
 171             sage: J = RealCartesianProductEJA(3)
 172             sage: p1 = J.one().characteristic_polynomial()
 173             sage: q1 = J.zero().characteristic_polynomial()
 174             sage: e0,e1,e2 = J.gens()
 175             sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3
 176             sage: p2 = A.one().characteristic_polynomial()
 177             sage: q2 = A.zero().characteristic_polynomial()
 178             sage: p1 == p2
 179             True
 180             sage: q1 == q2
 181             True
 182
 183         """
 184         p = self.parent().characteristic_polynomial()
 185         return p(*self.to_vector())
 186
 187
 188     def inner_product(self, other):
 189         """
 190         Return the parent algebra's inner product of myself and ``other``.
 191
 192         SETUP::
 193
 194             sage: from mjo.eja.eja_algebra import (
 195             ....:   ComplexHermitianEJA,
 196             ....:   JordanSpinEJA,
 197             ....:   QuaternionHermitianEJA,
 198             ....:   RealSymmetricEJA,
 199             ....:   random_eja)
 200
 201         EXAMPLES:
 202
 203         The inner product in the Jordan spin algebra is the usual
 204         inner product on `R^n` (this example only works because the
 205         basis for the Jordan algebra is the standard basis in `R^n`)::
 206
 207             sage: J = JordanSpinEJA(3)
 208             sage: x = vector(QQ,[1,2,3])
 209             sage: y = vector(QQ,[4,5,6])
 210             sage: x.inner_product(y)
 211             32
 212             sage: J.from_vector(x).inner_product(J.from_vector(y))
 213             32
 214
 215         The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
 216         multiplication is the usual matrix multiplication in `S^n`,
 217         so the inner product of the identity matrix with itself
 218         should be the `n`::
 219
 220             sage: J = RealSymmetricEJA(3)
 221             sage: J.one().inner_product(J.one())
 222             3
 223
 224         Likewise, the inner product on `C^n` is `<X,Y> =
 225         Re(trace(X*Y))`, where we must necessarily take the real
 226         part because the product of Hermitian matrices may not be
 227         Hermitian::
 228
 229             sage: J = ComplexHermitianEJA(3)
 230             sage: J.one().inner_product(J.one())
 231             3
 232
 233         Ditto for the quaternions::
 234
 235             sage: J = QuaternionHermitianEJA(3)
 236             sage: J.one().inner_product(J.one())
 237             3
 238
 239         TESTS:
 240
 241         Ensure that we can always compute an inner product, and that
 242         it gives us back a real number::
 243
 244             sage: set_random_seed()
 245             sage: J = random_eja()
 246             sage: x = J.random_element()
 247             sage: y = J.random_element()
 248             sage: x.inner_product(y) in RR
 249             True
 250
 251         """
 252         P = self.parent()
 253         if not other in P:
 254             raise TypeError("'other' must live in the same algebra")
 255
 256         return P.inner_product(self, other)
 257
 258
 259     def operator_commutes_with(self, other):
 260         """
 261         Return whether or not this element operator-commutes
 262         with ``other``.
 263
 264         SETUP::
 265
 266             sage: from mjo.eja.eja_algebra import random_eja
 267
 268         EXAMPLES:
 269
 270         The definition of a Jordan algebra says that any element
 271         operator-commutes with its square::
 272
 273             sage: set_random_seed()
 274             sage: x = random_eja().random_element()
 275             sage: x.operator_commutes_with(x^2)
 276             True
 277
 278         TESTS:
 279
 280         Test Lemma 1 from Chapter III of Koecher::
 281
 282             sage: set_random_seed()
 283             sage: J = random_eja()
 284             sage: u = J.random_element()
 285             sage: v = J.random_element()
 286             sage: lhs = u.operator_commutes_with(u*v)
 287             sage: rhs = v.operator_commutes_with(u^2)
 288             sage: lhs == rhs
 289             True
 290
 291         Test the first polarization identity from my notes, Koecher
 292         Chapter III, or from Baes (2.3)::
 293
 294             sage: set_random_seed()
 295             sage: J = random_eja()
 296             sage: x = J.random_element()
 297             sage: y = J.random_element()
 298             sage: Lx = x.operator()
 299             sage: Ly = y.operator()
 300             sage: Lxx = (x*x).operator()
 301             sage: Lxy = (x*y).operator()
 302             sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
 303             True
 304
 305         Test the second polarization identity from my notes or from
 306         Baes (2.4)::
 307
 308             sage: set_random_seed()
 309             sage: J = random_eja()
 310             sage: x = J.random_element()
 311             sage: y = J.random_element()
 312             sage: z = J.random_element()
 313             sage: Lx = x.operator()
 314             sage: Ly = y.operator()
 315             sage: Lz = z.operator()
 316             sage: Lzy = (z*y).operator()
 317             sage: Lxy = (x*y).operator()
 318             sage: Lxz = (x*z).operator()
 319             sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
 320             True
 321
 322         Test the third polarization identity from my notes or from
 323         Baes (2.5)::
 324
 325             sage: set_random_seed()
 326             sage: J = random_eja()
 327             sage: u = J.random_element()
 328             sage: y = J.random_element()
 329             sage: z = J.random_element()
 330             sage: Lu = u.operator()
 331             sage: Ly = y.operator()
 332             sage: Lz = z.operator()
 333             sage: Lzy = (z*y).operator()
 334             sage: Luy = (u*y).operator()
 335             sage: Luz = (u*z).operator()
 336             sage: Luyz = (u*(y*z)).operator()
 337             sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
 338             sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
 339             sage: bool(lhs == rhs)
 340             True
 341
 342         """
 343         if not other in self.parent():
 344             raise TypeError("'other' must live in the same algebra")
 345
 346         A = self.operator()
 347         B = other.operator()
 348         return (A*B == B*A)
 349
 350
 351     def det(self):
 352         """
 353         Return my determinant, the product of my eigenvalues.
 354
 355         SETUP::
 356
 357             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 358             ....:                                  random_eja)
 359
 360         EXAMPLES::
 361
 362             sage: J = JordanSpinEJA(2)
 363             sage: e0,e1 = J.gens()
 364             sage: x = sum( J.gens() )
 365             sage: x.det()
 366             0
 367
 368         ::
 369
 370             sage: J = JordanSpinEJA(3)
 371             sage: e0,e1,e2 = J.gens()
 372             sage: x = sum( J.gens() )
 373             sage: x.det()
 374             -1
 375
 376         TESTS:
 377
 378         An element is invertible if and only if its determinant is
 379         non-zero::
 380
 381             sage: set_random_seed()
 382             sage: x = random_eja().random_element()
 383             sage: x.is_invertible() == (x.det() != 0)
 384             True
 385
 386         Ensure that the determinant is multiplicative on an associative
 387         subalgebra as in Faraut and Koranyi's Proposition II.2.2::
 388
 389             sage: set_random_seed()
 390             sage: J = random_eja().random_element().subalgebra_generated_by()
 391             sage: x = J.random_element()
 392             sage: y = J.random_element()
 393             sage: (x*y).det() == x.det()*y.det()
 394             True
 395
 396         """
 397         P = self.parent()
 398         r = P.rank()
 399         p = P._charpoly_coeff(0)
 400         # The _charpoly_coeff function already adds the factor of
 401         # -1 to ensure that _charpoly_coeff(0) is really what
 402         # appears in front of t^{0} in the charpoly. However,
 403         # we want (-1)^r times THAT for the determinant.
 404         return ((-1)**r)*p(*self.to_vector())
 405
 406
 407     def inverse(self):
 408         """
 409         Return the Jordan-multiplicative inverse of this element.
 410
 411         ALGORITHM:
 412
 413         We appeal to the quadratic representation as in Koecher's
 414         Theorem 12 in Chapter III, Section 5.
 415
 416         SETUP::
 417
 418             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 419             ....:                                  random_eja)
 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 not x.is_invertible():
 431             ....:     x = J.random_element()
 432             sage: x_vec = x.to_vector()
 433             sage: x0 = x_vec[0]
 434             sage: x_bar = x_vec[1:]
 435             sage: coeff = ~(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.from_vector(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::
 451
 452             sage: set_random_seed()
 453             sage: J = random_eja()
 454             sage: x = J.random_element()
 455             sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
 456             True
 457
 458         The inverse of the inverse is what we started with::
 459
 460             sage: set_random_seed()
 461             sage: J = random_eja()
 462             sage: x = J.random_element()
 463             sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
 464             True
 465
 466         The zero element is never invertible::
 467
 468             sage: set_random_seed()
 469             sage: J = random_eja().zero().inverse()
 470             Traceback (most recent call last):
 471             ...
 472             ValueError: element is not invertible
 473
 474         """
 475         if not self.is_invertible():
 476             raise ValueError("element is not invertible")
 477
 478         return (~self.quadratic_representation())(self)
 479
 480
 481     def is_invertible(self):
 482         """
 483         Return whether or not this element is invertible.
 484
 485         ALGORITHM:
 486
 487         The usual way to do this is to check if the determinant is
 488         zero, but we need the characteristic polynomial for the
 489         determinant. The minimal polynomial is a lot easier to get,
 490         so we use Corollary 2 in Chapter V of Koecher to check
 491         whether or not the paren't algebra's zero element is a root
 492         of this element's minimal polynomial.
 493
 494         Beware that we can't use the superclass method, because it
 495         relies on the algebra being associative.
 496
 497         SETUP::
 498
 499             sage: from mjo.eja.eja_algebra import random_eja
 500
 501         TESTS:
 502
 503         The identity element is always invertible::
 504
 505             sage: set_random_seed()
 506             sage: J = random_eja()
 507             sage: J.one().is_invertible()
 508             True
 509
 510         The zero element is never invertible in a non-trivial algebra::
 511
 512             sage: set_random_seed()
 513             sage: J = random_eja()
 514             sage: (not J.is_trivial()) and J.zero().is_invertible()
 515             False
 516
 517         """
 518         if self.is_zero():
 519             if self.parent().is_trivial():
 520                 return True
 521             else:
 522                 return False
 523
 524         # In fact, we only need to know if the constant term is non-zero,
 525         # so we can pass in the field's zero element instead.
 526         zero = self.base_ring().zero()
 527         p = self.minimal_polynomial()
 528         return not (p(zero) == zero)
 529
 530
 531     def is_nilpotent(self):
 532         """
 533         Return whether or not some power of this element is zero.
 534
 535         ALGORITHM:
 536
 537         We use Theorem 5 in Chapter III of Koecher, which says that
 538         an element ``x`` is nilpotent if and only if ``x.operator()``
 539         is nilpotent. And it is a basic fact of linear algebra that
 540         an operator on an `n`-dimensional space is nilpotent if and
 541         only if, when raised to the `n`th power, it equals the zero
 542         operator (for example, see Axler Corollary 8.8).
 543
 544         SETUP::
 545
 546             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 547             ....:                                  random_eja)
 548
 549         EXAMPLES::
 550
 551             sage: J = JordanSpinEJA(3)
 552             sage: x = sum(J.gens())
 553             sage: x.is_nilpotent()
 554             False
 555
 556         TESTS:
 557
 558         The identity element is never nilpotent::
 559
 560             sage: set_random_seed()
 561             sage: random_eja().one().is_nilpotent()
 562             False
 563
 564         The additive identity is always nilpotent::
 565
 566             sage: set_random_seed()
 567             sage: random_eja().zero().is_nilpotent()
 568             True
 569
 570         """
 571         P = self.parent()
 572         zero_operator = P.zero().operator()
 573         return self.operator()**P.dimension() == zero_operator
 574
 575
 576     def is_regular(self):
 577         """
 578         Return whether or not this is a regular element.
 579
 580         SETUP::
 581
 582             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 583             ....:                                  random_eja)
 584
 585         EXAMPLES:
 586
 587         The identity element always has degree one, but any element
 588         linearly-independent from it is regular::
 589
 590             sage: J = JordanSpinEJA(5)
 591             sage: J.one().is_regular()
 592             False
 593             sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
 594             sage: for x in J.gens():
 595             ....:     (J.one() + x).is_regular()
 596             False
 597             True
 598             True
 599             True
 600             True
 601
 602         TESTS:
 603
 604         The zero element should never be regular, unless the parent
 605         algebra has dimension one::
 606
 607             sage: set_random_seed()
 608             sage: J = random_eja()
 609             sage: J.dimension() == 1 or not J.zero().is_regular()
 610             True
 611
 612         The unit element isn't regular unless the algebra happens to
 613         consist of only its scalar multiples::
 614
 615             sage: set_random_seed()
 616             sage: J = random_eja()
 617             sage: J.dimension() == 1 or not J.one().is_regular()
 618             True
 619
 620         """
 621         return self.degree() == self.parent().rank()
 622
 623
 624     def degree(self):
 625         """
 626         Return the degree of this element, which is defined to be
 627         the degree of its minimal polynomial.
 628
 629         ALGORITHM:
 630
 631         For now, we skip the messy minimal polynomial computation
 632         and instead return the dimension of the vector space spanned
 633         by the powers of this element. The latter is a bit more
 634         straightforward to compute.
 635
 636         SETUP::
 637
 638             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 639             ....:                                  random_eja)
 640
 641         EXAMPLES::
 642
 643             sage: J = JordanSpinEJA(4)
 644             sage: J.one().degree()
 645             1
 646             sage: e0,e1,e2,e3 = J.gens()
 647             sage: (e0 - e1).degree()
 648             2
 649
 650         In the spin factor algebra (of rank two), all elements that
 651         aren't multiples of the identity are regular::
 652
 653             sage: set_random_seed()
 654             sage: n = ZZ.random_element(1,10)
 655             sage: J = JordanSpinEJA(n)
 656             sage: x = J.random_element()
 657             sage: x == x.coefficient(0)*J.one() or x.degree() == 2
 658             True
 659
 660         TESTS:
 661
 662         The zero and unit elements are both of degree one::
 663
 664             sage: set_random_seed()
 665             sage: J = random_eja()
 666             sage: J.zero().degree()
 667             1
 668             sage: J.one().degree()
 669             1
 670
 671         Our implementation agrees with the definition::
 672
 673             sage: set_random_seed()
 674             sage: x = random_eja().random_element()
 675             sage: x.degree() == x.minimal_polynomial().degree()
 676             True
 677
 678         """
 679         if self.is_zero() and not self.parent().is_trivial():
 680             # The minimal polynomial of zero in a nontrivial algebra
 681             # is "t"; in a trivial algebra it's "1" by convention
 682             # (it's an empty product).
 683             return 1
 684         return self.subalgebra_generated_by().dimension()
 685
 686
 687     def left_matrix(self):
 688         """
 689         Our parent class defines ``left_matrix`` and ``matrix``
 690         methods whose names are misleading. We don't want them.
 691         """
 692         raise NotImplementedError("use operator().matrix() instead")
 693
 694     matrix = left_matrix
 695
 696
 697     def minimal_polynomial(self):
 698         """
 699         Return the minimal polynomial of this element,
 700         as a function of the variable `t`.
 701
 702         ALGORITHM:
 703
 704         We restrict ourselves to the associative subalgebra
 705         generated by this element, and then return the minimal
 706         polynomial of this element's operator matrix (in that
 707         subalgebra). This works by Baes Proposition 2.3.16.
 708
 709         SETUP::
 710
 711             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 712             ....:                                  random_eja)
 713
 714         TESTS:
 715
 716         The minimal polynomial of the identity and zero elements are
 717         always the same::
 718
 719             sage: set_random_seed()
 720             sage: J = random_eja()
 721             sage: J.one().minimal_polynomial()
 722             t - 1
 723             sage: J.zero().minimal_polynomial()
 724             t
 725
 726         The degree of an element is (by one definition) the degree
 727         of its minimal polynomial::
 728
 729             sage: set_random_seed()
 730             sage: x = random_eja().random_element()
 731             sage: x.degree() == x.minimal_polynomial().degree()
 732             True
 733
 734         The minimal polynomial and the characteristic polynomial coincide
 735         and are known (see Alizadeh, Example 11.11) for all elements of
 736         the spin factor algebra that aren't scalar multiples of the
 737         identity::
 738
 739             sage: set_random_seed()
 740             sage: n = ZZ.random_element(2,10)
 741             sage: J = JordanSpinEJA(n)
 742             sage: y = J.random_element()
 743             sage: while y == y.coefficient(0)*J.one():
 744             ....:     y = J.random_element()
 745             sage: y0 = y.to_vector()[0]
 746             sage: y_bar = y.to_vector()[1:]
 747             sage: actual = y.minimal_polynomial()
 748             sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
 749             sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
 750             sage: bool(actual == expected)
 751             True
 752
 753         The minimal polynomial should always kill its element::
 754
 755             sage: set_random_seed()
 756             sage: x = random_eja().random_element()
 757             sage: p = x.minimal_polynomial()
 758             sage: x.apply_univariate_polynomial(p)
 759             0
 760
 761         """
 762         if self.is_zero():
 763             # We would generate a zero-dimensional subalgebra
 764             # where the minimal polynomial would be constant.
 765             # That might be correct, but only if *this* algebra
 766             # is trivial too.
 767             if not self.parent().is_trivial():
 768                 # Pretty sure we know what the minimal polynomial of
 769                 # the zero operator is going to be. This ensures
 770                 # consistency of e.g. the polynomial variable returned
 771                 # in the "normal" case without us having to think about it.
 772                 return self.operator().minimal_polynomial()
 773
 774         A = self.subalgebra_generated_by()
 775         return A(self).operator().minimal_polynomial()
 776
 777
 778
 779     def natural_representation(self):
 780         """
 781         Return a more-natural representation of this element.
 782
 783         Every finite-dimensional Euclidean Jordan Algebra is a
 784         direct sum of five simple algebras, four of which comprise
 785         Hermitian matrices. This method returns the original
 786         "natural" representation of this element as a Hermitian
 787         matrix, if it has one. If not, you get the usual representation.
 788
 789         SETUP::
 790
 791             sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
 792             ....:                                  QuaternionHermitianEJA)
 793
 794         EXAMPLES::
 795
 796             sage: J = ComplexHermitianEJA(3)
 797             sage: J.one()
 798             e0 + e3 + e8
 799             sage: J.one().natural_representation()
 800             [1 0 0 0 0 0]
 801             [0 1 0 0 0 0]
 802             [0 0 1 0 0 0]
 803             [0 0 0 1 0 0]
 804             [0 0 0 0 1 0]
 805             [0 0 0 0 0 1]
 806
 807         ::
 808
 809             sage: J = QuaternionHermitianEJA(3)
 810             sage: J.one()
 811             e0 + e5 + e14
 812             sage: J.one().natural_representation()
 813             [1 0 0 0 0 0 0 0 0 0 0 0]
 814             [0 1 0 0 0 0 0 0 0 0 0 0]
 815             [0 0 1 0 0 0 0 0 0 0 0 0]
 816             [0 0 0 1 0 0 0 0 0 0 0 0]
 817             [0 0 0 0 1 0 0 0 0 0 0 0]
 818             [0 0 0 0 0 1 0 0 0 0 0 0]
 819             [0 0 0 0 0 0 1 0 0 0 0 0]
 820             [0 0 0 0 0 0 0 1 0 0 0 0]
 821             [0 0 0 0 0 0 0 0 1 0 0 0]
 822             [0 0 0 0 0 0 0 0 0 1 0 0]
 823             [0 0 0 0 0 0 0 0 0 0 1 0]
 824             [0 0 0 0 0 0 0 0 0 0 0 1]
 825
 826         """
 827         B = self.parent().natural_basis()
 828         W = self.parent().natural_basis_space()
 829         return W.linear_combination(zip(B,self.to_vector()))
 830
 831
 832     def operator(self):
 833         """
 834         Return the left-multiplication-by-this-element
 835         operator on the ambient algebra.
 836
 837         SETUP::
 838
 839             sage: from mjo.eja.eja_algebra import random_eja
 840
 841         TESTS::
 842
 843             sage: set_random_seed()
 844             sage: J = random_eja()
 845             sage: x = J.random_element()
 846             sage: y = J.random_element()
 847             sage: x.operator()(y) == x*y
 848             True
 849             sage: y.operator()(x) == x*y
 850             True
 851
 852         """
 853         P = self.parent()
 854         left_mult_by_self = lambda y: self*y
 855         L = P.module_morphism(function=left_mult_by_self, codomain=P)
 856         return FiniteDimensionalEuclideanJordanAlgebraOperator(
 857                  P,
 858                  P,
 859                  L.matrix() )
 860
 861
 862     def quadratic_representation(self, other=None):
 863         """
 864         Return the quadratic representation of this element.
 865
 866         SETUP::
 867
 868             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 869             ....:                                  random_eja)
 870
 871         EXAMPLES:
 872
 873         The explicit form in the spin factor algebra is given by
 874         Alizadeh's Example 11.12::
 875
 876             sage: set_random_seed()
 877             sage: n = ZZ.random_element(1,10)
 878             sage: J = JordanSpinEJA(n)
 879             sage: x = J.random_element()
 880             sage: x_vec = x.to_vector()
 881             sage: x0 = x_vec[0]
 882             sage: x_bar = x_vec[1:]
 883             sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
 884             sage: B = 2*x0*x_bar.row()
 885             sage: C = 2*x0*x_bar.column()
 886             sage: D = matrix.identity(QQ, n-1)
 887             sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
 888             sage: D = D + 2*x_bar.tensor_product(x_bar)
 889             sage: Q = matrix.block(2,2,[A,B,C,D])
 890             sage: Q == x.quadratic_representation().matrix()
 891             True
 892
 893         Test all of the properties from Theorem 11.2 in Alizadeh::
 894
 895             sage: set_random_seed()
 896             sage: J = random_eja()
 897             sage: x = J.random_element()
 898             sage: y = J.random_element()
 899             sage: Lx = x.operator()
 900             sage: Lxx = (x*x).operator()
 901             sage: Qx = x.quadratic_representation()
 902             sage: Qy = y.quadratic_representation()
 903             sage: Qxy = x.quadratic_representation(y)
 904             sage: Qex = J.one().quadratic_representation(x)
 905             sage: n = ZZ.random_element(10)
 906             sage: Qxn = (x^n).quadratic_representation()
 907
 908         Property 1:
 909
 910             sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
 911             True
 912
 913         Property 2 (multiply on the right for :trac:`28272`):
 914
 915             sage: alpha = QQ.random_element()
 916             sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
 917             True
 918
 919         Property 3:
 920
 921             sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
 922             True
 923
 924             sage: not x.is_invertible() or (
 925             ....:   ~Qx
 926             ....:   ==
 927             ....:   x.inverse().quadratic_representation() )
 928             True
 929
 930             sage: Qxy(J.one()) == x*y
 931             True
 932
 933         Property 4:
 934
 935             sage: not x.is_invertible() or (
 936             ....:   x.quadratic_representation(x.inverse())*Qx
 937             ....:   == Qx*x.quadratic_representation(x.inverse()) )
 938             True
 939
 940             sage: not x.is_invertible() or (
 941             ....:   x.quadratic_representation(x.inverse())*Qx
 942             ....:   ==
 943             ....:   2*x.operator()*Qex - Qx )
 944             True
 945
 946             sage: 2*x.operator()*Qex - Qx == Lxx
 947             True
 948
 949         Property 5:
 950
 951             sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
 952             True
 953
 954         Property 6:
 955
 956             sage: Qxn == (Qx)^n
 957             True
 958
 959         Property 7:
 960
 961             sage: not x.is_invertible() or (
 962             ....:   Qx*x.inverse().operator() == Lx )
 963             True
 964
 965         Property 8:
 966
 967             sage: not x.operator_commutes_with(y) or (
 968             ....:   Qx(y)^n == Qxn(y^n) )
 969             True
 970
 971         """
 972         if other is None:
 973             other=self
 974         elif not other in self.parent():
 975             raise TypeError("'other' must live in the same algebra")
 976
 977         L = self.operator()
 978         M = other.operator()
 979         return ( L*M + M*L - (self*other).operator() )
 980
 981
 982
 983
 984     def subalgebra_generated_by(self):
 985         """
 986         Return the associative subalgebra of the parent EJA generated
 987         by this element.
 988
 989         SETUP::
 990
 991             sage: from mjo.eja.eja_algebra import random_eja
 992
 993         TESTS:
 994
 995         This subalgebra, being composed of only powers, is associative::
 996
 997             sage: set_random_seed()
 998             sage: x0 = random_eja().random_element()
 999             sage: A = x0.subalgebra_generated_by()
1000             sage: x = A.random_element()
1001             sage: y = A.random_element()
1002             sage: z = A.random_element()
1003             sage: (x*y)*z == x*(y*z)
1004             True
1005
1006         Squaring in the subalgebra should work the same as in
1007         the superalgebra::
1008
1009             sage: set_random_seed()
1010             sage: x = random_eja().random_element()
1011             sage: A = x.subalgebra_generated_by()
1012             sage: A(x^2) == A(x)*A(x)
1013             True
1014
1015         The subalgebra generated by the zero element is trivial::
1016
1017             sage: set_random_seed()
1018             sage: A = random_eja().zero().subalgebra_generated_by()
1019             sage: A
1020             Euclidean Jordan algebra of dimension 0 over Rational Field
1021             sage: A.one()
1022             0
1023
1024         """
1025         return FiniteDimensionalEuclideanJordanElementSubalgebra(self)
1026
1027
1028     def subalgebra_idempotent(self):
1029         """
1030         Find an idempotent in the associative subalgebra I generate
1031         using Proposition 2.3.5 in Baes.
1032
1033         SETUP::
1034
1035             sage: from mjo.eja.eja_algebra import random_eja
1036
1037         TESTS::
1038
1039             sage: set_random_seed()
1040             sage: J = random_eja()
1041             sage: x = J.random_element()
1042             sage: while x.is_nilpotent():
1043             ....:     x = J.random_element()
1044             sage: c = x.subalgebra_idempotent()
1045             sage: c^2 == c
1046             True
1047
1048         """
1049         if self.is_nilpotent():
1050             raise ValueError("this only works with non-nilpotent elements!")
1051
1052         J = self.subalgebra_generated_by()
1053         u = J(self)
1054
1055         # The image of the matrix of left-u^m-multiplication
1056         # will be minimal for some natural number s...
1057         s = 0
1058         minimal_dim = J.dimension()
1059         for i in xrange(1, minimal_dim):
1060             this_dim = (u**i).operator().matrix().image().dimension()
1061             if this_dim < minimal_dim:
1062                 minimal_dim = this_dim
1063                 s = i
1064
1065         # Now minimal_matrix should correspond to the smallest
1066         # non-zero subspace in Baes's (or really, Koecher's)
1067         # proposition.
1068         #
1069         # However, we need to restrict the matrix to work on the
1070         # subspace... or do we? Can't we just solve, knowing that
1071         # A(c) = u^(s+1) should have a solution in the big space,
1072         # too?
1073         #
1074         # Beware, solve_right() means that we're using COLUMN vectors.
1075         # Our FiniteDimensionalAlgebraElement superclass uses rows.
1076         u_next = u**(s+1)
1077         A = u_next.operator().matrix()
1078         c = J.from_vector(A.solve_right(u_next.to_vector()))
1079
1080         # Now c is the idempotent we want, but it still lives in the subalgebra.
1081         return c.superalgebra_element()
1082
1083
1084     def trace(self):
1085         """
1086         Return my trace, the sum of my eigenvalues.
1087
1088         SETUP::
1089
1090             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1091             ....:                                  RealCartesianProductEJA,
1092             ....:                                  random_eja)
1093
1094         EXAMPLES::
1095
1096             sage: J = JordanSpinEJA(3)
1097             sage: x = sum(J.gens())
1098             sage: x.trace()
1099             2
1100
1101         ::
1102
1103             sage: J = RealCartesianProductEJA(5)
1104             sage: J.one().trace()
1105             5
1106
1107         TESTS:
1108
1109         The trace of an element is a real number::
1110
1111             sage: set_random_seed()
1112             sage: J = random_eja()
1113             sage: J.random_element().trace() in J.base_ring()
1114             True
1115
1116         """
1117         P = self.parent()
1118         r = P.rank()
1119         p = P._charpoly_coeff(r-1)
1120         # The _charpoly_coeff function already adds the factor of
1121         # -1 to ensure that _charpoly_coeff(r-1) is really what
1122         # appears in front of t^{r-1} in the charpoly. However,
1123         # we want the negative of THAT for the trace.
1124         return -p(*self.to_vector())
1125
1126
1127     def trace_inner_product(self, other):
1128         """
1129         Return the trace inner product of myself and ``other``.
1130
1131         SETUP::
1132
1133             sage: from mjo.eja.eja_algebra import random_eja
1134
1135         TESTS:
1136
1137         The trace inner product is commutative::
1138
1139             sage: set_random_seed()
1140             sage: J = random_eja()
1141             sage: x = J.random_element(); y = J.random_element()
1142             sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1143             True
1144
1145         The trace inner product is bilinear::
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: z = J.random_element()
1152             sage: a = QQ.random_element();
1153             sage: actual = (a*(x+z)).trace_inner_product(y)
1154             sage: expected = ( a*x.trace_inner_product(y) +
1155             ....:              a*z.trace_inner_product(y) )
1156             sage: actual == expected
1157             True
1158             sage: actual = x.trace_inner_product(a*(y+z))
1159             sage: expected = ( a*x.trace_inner_product(y) +
1160             ....:              a*x.trace_inner_product(z) )
1161             sage: actual == expected
1162             True
1163
1164         The trace inner product satisfies the compatibility
1165         condition in the definition of a Euclidean Jordan algebra::
1166
1167             sage: set_random_seed()
1168             sage: J = random_eja()
1169             sage: x = J.random_element()
1170             sage: y = J.random_element()
1171             sage: z = J.random_element()
1172             sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1173             True
1174
1175         """
1176         if not other in self.parent():
1177             raise TypeError("'other' must live in the same algebra")
1178
1179         return (self*other).trace()