mjo/eja/eja_element.py

   1 # -*- coding: utf-8 -*-
   2
   3 from itertools import izip
   4
   5 from sage.matrix.constructor import matrix
   6 from sage.modules.free_module import VectorSpace
   7 from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
   8
   9 # TODO: make this unnecessary somehow.
  10 from sage.misc.lazy_import import lazy_import
  11 lazy_import('mjo.eja.eja_algebra', 'FiniteDimensionalEuclideanJordanAlgebra')
  12 lazy_import('mjo.eja.eja_subalgebra',
  13             'FiniteDimensionalEuclideanJordanElementSubalgebra')
  14 from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
  15 from mjo.eja.eja_utils import _mat2vec
  16
  17 class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
  18     """
  19     An element of a Euclidean Jordan algebra.
  20     """
  21
  22     def __dir__(self):
  23         """
  24         Oh man, I should not be doing this. This hides the "disabled"
  25         methods ``left_matrix`` and ``matrix`` from introspection;
  26         in particular it removes them from tab-completion.
  27         """
  28         return filter(lambda s: s not in ['left_matrix', 'matrix'],
  29                       dir(self.__class__) )
  30
  31
  32
  33
  34     def __pow__(self, n):
  35         """
  36         Return ``self`` raised to the power ``n``.
  37
  38         Jordan algebras are always power-associative; see for
  39         example Faraut and Korányi, Proposition II.1.2 (ii).
  40
  41         We have to override this because our superclass uses row
  42         vectors instead of column vectors! We, on the other hand,
  43         assume column vectors everywhere.
  44
  45         SETUP::
  46
  47             sage: from mjo.eja.eja_algebra import random_eja
  48
  49         TESTS:
  50
  51         The definition of `x^2` is the unambiguous `x*x`::
  52
  53             sage: set_random_seed()
  54             sage: x = random_eja().random_element()
  55             sage: x*x == (x^2)
  56             True
  57
  58         A few examples of power-associativity::
  59
  60             sage: set_random_seed()
  61             sage: x = random_eja().random_element()
  62             sage: x*(x*x)*(x*x) == x^5
  63             True
  64             sage: (x*x)*(x*x*x) == x^5
  65             True
  66
  67         We also know that powers operator-commute (Koecher, Chapter
  68         III, Corollary 1)::
  69
  70             sage: set_random_seed()
  71             sage: x = random_eja().random_element()
  72             sage: m = ZZ.random_element(0,10)
  73             sage: n = ZZ.random_element(0,10)
  74             sage: Lxm = (x^m).operator()
  75             sage: Lxn = (x^n).operator()
  76             sage: Lxm*Lxn == Lxn*Lxm
  77             True
  78
  79         """
  80         if n == 0:
  81             return self.parent().one()
  82         elif n == 1:
  83             return self
  84         else:
  85             return (self**(n-1))*self
  86
  87
  88     def apply_univariate_polynomial(self, p):
  89         """
  90         Apply the univariate polynomial ``p`` to this element.
  91
  92         A priori, SageMath won't allow us to apply a univariate
  93         polynomial to an element of an EJA, because we don't know
  94         that EJAs are rings (they are usually not associative). Of
  95         course, we know that EJAs are power-associative, so the
  96         operation is ultimately kosher. This function sidesteps
  97         the CAS to get the answer we want and expect.
  98
  99         SETUP::
 100
 101             sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
 102             ....:                                  random_eja)
 103
 104         EXAMPLES::
 105
 106             sage: R = PolynomialRing(QQ, 't')
 107             sage: t = R.gen(0)
 108             sage: p = t^4 - t^3 + 5*t - 2
 109             sage: J = RealCartesianProductEJA(5)
 110             sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
 111             True
 112
 113         TESTS:
 114
 115         We should always get back an element of the algebra::
 116
 117             sage: set_random_seed()
 118             sage: p = PolynomialRing(QQ, 't').random_element()
 119             sage: J = random_eja()
 120             sage: x = J.random_element()
 121             sage: x.apply_univariate_polynomial(p) in J
 122             True
 123
 124         """
 125         if len(p.variables()) > 1:
 126             raise ValueError("not a univariate polynomial")
 127         P = self.parent()
 128         R = P.base_ring()
 129         # Convert the coeficcients to the parent's base ring,
 130         # because a priori they might live in an (unnecessarily)
 131         # larger ring for which P.sum() would fail below.
 132         cs = [ R(c) for c in p.coefficients(sparse=False) ]
 133         return P.sum( cs[k]*(self**k) for k in range(len(cs)) )
 134
 135
 136     def characteristic_polynomial(self):
 137         """
 138         Return the characteristic polynomial of this element.
 139
 140         SETUP::
 141
 142             sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
 143
 144         EXAMPLES:
 145
 146         The rank of `R^3` is three, and the minimal polynomial of
 147         the identity element is `(t-1)` from which it follows that
 148         the characteristic polynomial should be `(t-1)^3`::
 149
 150             sage: J = RealCartesianProductEJA(3)
 151             sage: J.one().characteristic_polynomial()
 152             t^3 - 3*t^2 + 3*t - 1
 153
 154         Likewise, the characteristic of the zero element in the
 155         rank-three algebra `R^{n}` should be `t^{3}`::
 156
 157             sage: J = RealCartesianProductEJA(3)
 158             sage: J.zero().characteristic_polynomial()
 159             t^3
 160
 161         TESTS:
 162
 163         The characteristic polynomial of an element should evaluate
 164         to zero on that element::
 165
 166             sage: set_random_seed()
 167             sage: x = RealCartesianProductEJA(3).random_element()
 168             sage: p = x.characteristic_polynomial()
 169             sage: x.apply_univariate_polynomial(p)
 170             0
 171
 172         The characteristic polynomials of the zero and unit elements
 173         should be what we think they are in a subalgebra, too::
 174
 175             sage: J = RealCartesianProductEJA(3)
 176             sage: p1 = J.one().characteristic_polynomial()
 177             sage: q1 = J.zero().characteristic_polynomial()
 178             sage: e0,e1,e2 = J.gens()
 179             sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3
 180             sage: p2 = A.one().characteristic_polynomial()
 181             sage: q2 = A.zero().characteristic_polynomial()
 182             sage: p1 == p2
 183             True
 184             sage: q1 == q2
 185             True
 186
 187         """
 188         p = self.parent().characteristic_polynomial()
 189         return p(*self.to_vector())
 190
 191
 192     def inner_product(self, other):
 193         """
 194         Return the parent algebra's inner product of myself and ``other``.
 195
 196         SETUP::
 197
 198             sage: from mjo.eja.eja_algebra import (
 199             ....:   ComplexHermitianEJA,
 200             ....:   JordanSpinEJA,
 201             ....:   QuaternionHermitianEJA,
 202             ....:   RealSymmetricEJA,
 203             ....:   random_eja)
 204
 205         EXAMPLES:
 206
 207         The inner product in the Jordan spin algebra is the usual
 208         inner product on `R^n` (this example only works because the
 209         basis for the Jordan algebra is the standard basis in `R^n`)::
 210
 211             sage: J = JordanSpinEJA(3)
 212             sage: x = vector(QQ,[1,2,3])
 213             sage: y = vector(QQ,[4,5,6])
 214             sage: x.inner_product(y)
 215             32
 216             sage: J.from_vector(x).inner_product(J.from_vector(y))
 217             32
 218
 219         The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
 220         multiplication is the usual matrix multiplication in `S^n`,
 221         so the inner product of the identity matrix with itself
 222         should be the `n`::
 223
 224             sage: J = RealSymmetricEJA(3)
 225             sage: J.one().inner_product(J.one())
 226             3
 227
 228         Likewise, the inner product on `C^n` is `<X,Y> =
 229         Re(trace(X*Y))`, where we must necessarily take the real
 230         part because the product of Hermitian matrices may not be
 231         Hermitian::
 232
 233             sage: J = ComplexHermitianEJA(3)
 234             sage: J.one().inner_product(J.one())
 235             3
 236
 237         Ditto for the quaternions::
 238
 239             sage: J = QuaternionHermitianEJA(3)
 240             sage: J.one().inner_product(J.one())
 241             3
 242
 243         TESTS:
 244
 245         Ensure that we can always compute an inner product, and that
 246         it gives us back a real number::
 247
 248             sage: set_random_seed()
 249             sage: J = random_eja()
 250             sage: x,y = J.random_elements(2)
 251             sage: x.inner_product(y) in RLF
 252             True
 253
 254         """
 255         P = self.parent()
 256         if not other in P:
 257             raise TypeError("'other' must live in the same algebra")
 258
 259         return P.inner_product(self, other)
 260
 261
 262     def operator_commutes_with(self, other):
 263         """
 264         Return whether or not this element operator-commutes
 265         with ``other``.
 266
 267         SETUP::
 268
 269             sage: from mjo.eja.eja_algebra import random_eja
 270
 271         EXAMPLES:
 272
 273         The definition of a Jordan algebra says that any element
 274         operator-commutes with its square::
 275
 276             sage: set_random_seed()
 277             sage: x = random_eja().random_element()
 278             sage: x.operator_commutes_with(x^2)
 279             True
 280
 281         TESTS:
 282
 283         Test Lemma 1 from Chapter III of Koecher::
 284
 285             sage: set_random_seed()
 286             sage: u,v = random_eja().random_elements(2)
 287             sage: lhs = u.operator_commutes_with(u*v)
 288             sage: rhs = v.operator_commutes_with(u^2)
 289             sage: lhs == rhs
 290             True
 291
 292         Test the first polarization identity from my notes, Koecher
 293         Chapter III, or from Baes (2.3)::
 294
 295             sage: set_random_seed()
 296             sage: x,y = random_eja().random_elements(2)
 297             sage: Lx = x.operator()
 298             sage: Ly = y.operator()
 299             sage: Lxx = (x*x).operator()
 300             sage: Lxy = (x*y).operator()
 301             sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
 302             True
 303
 304         Test the second polarization identity from my notes or from
 305         Baes (2.4)::
 306
 307             sage: set_random_seed()
 308             sage: x,y,z = random_eja().random_elements(3)
 309             sage: Lx = x.operator()
 310             sage: Ly = y.operator()
 311             sage: Lz = z.operator()
 312             sage: Lzy = (z*y).operator()
 313             sage: Lxy = (x*y).operator()
 314             sage: Lxz = (x*z).operator()
 315             sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
 316             True
 317
 318         Test the third polarization identity from my notes or from
 319         Baes (2.5)::
 320
 321             sage: set_random_seed()
 322             sage: u,y,z = random_eja().random_elements(3)
 323             sage: Lu = u.operator()
 324             sage: Ly = y.operator()
 325             sage: Lz = z.operator()
 326             sage: Lzy = (z*y).operator()
 327             sage: Luy = (u*y).operator()
 328             sage: Luz = (u*z).operator()
 329             sage: Luyz = (u*(y*z)).operator()
 330             sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
 331             sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
 332             sage: bool(lhs == rhs)
 333             True
 334
 335         """
 336         if not other in self.parent():
 337             raise TypeError("'other' must live in the same algebra")
 338
 339         A = self.operator()
 340         B = other.operator()
 341         return (A*B == B*A)
 342
 343
 344     def det(self):
 345         """
 346         Return my determinant, the product of my eigenvalues.
 347
 348         SETUP::
 349
 350             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 351             ....:                                  random_eja)
 352
 353         EXAMPLES::
 354
 355             sage: J = JordanSpinEJA(2)
 356             sage: e0,e1 = J.gens()
 357             sage: x = sum( J.gens() )
 358             sage: x.det()
 359             0
 360
 361         ::
 362
 363             sage: J = JordanSpinEJA(3)
 364             sage: e0,e1,e2 = J.gens()
 365             sage: x = sum( J.gens() )
 366             sage: x.det()
 367             -1
 368
 369         TESTS:
 370
 371         An element is invertible if and only if its determinant is
 372         non-zero::
 373
 374             sage: set_random_seed()
 375             sage: x = random_eja().random_element()
 376             sage: x.is_invertible() == (x.det() != 0)
 377             True
 378
 379         Ensure that the determinant is multiplicative on an associative
 380         subalgebra as in Faraut and Korányi's Proposition II.2.2::
 381
 382             sage: set_random_seed()
 383             sage: J = random_eja().random_element().subalgebra_generated_by()
 384             sage: x,y = J.random_elements(2)
 385             sage: (x*y).det() == x.det()*y.det()
 386             True
 387
 388         """
 389         P = self.parent()
 390         r = P.rank()
 391         p = P._charpoly_coeff(0)
 392         # The _charpoly_coeff function already adds the factor of
 393         # -1 to ensure that _charpoly_coeff(0) is really what
 394         # appears in front of t^{0} in the charpoly. However,
 395         # we want (-1)^r times THAT for the determinant.
 396         return ((-1)**r)*p(*self.to_vector())
 397
 398
 399     def inverse(self):
 400         """
 401         Return the Jordan-multiplicative inverse of this element.
 402
 403         ALGORITHM:
 404
 405         We appeal to the quadratic representation as in Koecher's
 406         Theorem 12 in Chapter III, Section 5.
 407
 408         SETUP::
 409
 410             sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
 411             ....:                                  JordanSpinEJA,
 412             ....:                                  random_eja)
 413
 414         EXAMPLES:
 415
 416         The inverse in the spin factor algebra is given in Alizadeh's
 417         Example 11.11::
 418
 419             sage: set_random_seed()
 420             sage: J = JordanSpinEJA.random_instance()
 421             sage: x = J.random_element()
 422             sage: while not x.is_invertible():
 423             ....:     x = J.random_element()
 424             sage: x_vec = x.to_vector()
 425             sage: x0 = x_vec[0]
 426             sage: x_bar = x_vec[1:]
 427             sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar))
 428             sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
 429             sage: x_inverse = coeff*inv_vec
 430             sage: x.inverse() == J.from_vector(x_inverse)
 431             True
 432
 433         Trying to invert a non-invertible element throws an error:
 434
 435             sage: JordanSpinEJA(3).zero().inverse()
 436             Traceback (most recent call last):
 437             ...
 438             ValueError: element is not invertible
 439
 440         TESTS:
 441
 442         The identity element is its own inverse::
 443
 444             sage: set_random_seed()
 445             sage: J = random_eja()
 446             sage: J.one().inverse() == J.one()
 447             True
 448
 449         If an element has an inverse, it acts like one::
 450
 451             sage: set_random_seed()
 452             sage: J = random_eja()
 453             sage: x = J.random_element()
 454             sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
 455             True
 456
 457         The inverse of the inverse is what we started with::
 458
 459             sage: set_random_seed()
 460             sage: J = random_eja()
 461             sage: x = J.random_element()
 462             sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
 463             True
 464
 465         Proposition II.2.3 in Faraut and Korányi says that the inverse
 466         of an element is the inverse of its left-multiplication operator
 467         applied to the algebra's identity, when that inverse exists::
 468
 469             sage: set_random_seed()
 470             sage: J = random_eja()
 471             sage: x = J.random_element()
 472             sage: (not x.operator().is_invertible()) or (
 473             ....:    x.operator().inverse()(J.one()) == x.inverse() )
 474             True
 475
 476         Proposition II.2.4 in Faraut and Korányi gives a formula for
 477         the inverse based on the characteristic polynomial and the
 478         Cayley-Hamilton theorem for Euclidean Jordan algebras::
 479
 480             sage: set_random_seed()
 481             sage: J = ComplexHermitianEJA(3)
 482             sage: x = J.random_element()
 483             sage: while not x.is_invertible():
 484             ....:     x = J.random_element()
 485             sage: r = J.rank()
 486             sage: a = x.characteristic_polynomial().coefficients(sparse=False)
 487             sage: expected  = (-1)^(r+1)/x.det()
 488             sage: expected *= sum( a[i+1]*x^i for i in range(r) )
 489             sage: x.inverse() == expected
 490             True
 491
 492         """
 493         if not self.is_invertible():
 494             raise ValueError("element is not invertible")
 495
 496         return (~self.quadratic_representation())(self)
 497
 498
 499     def is_invertible(self):
 500         """
 501         Return whether or not this element is invertible.
 502
 503         ALGORITHM:
 504
 505         The usual way to do this is to check if the determinant is
 506         zero, but we need the characteristic polynomial for the
 507         determinant. The minimal polynomial is a lot easier to get,
 508         so we use Corollary 2 in Chapter V of Koecher to check
 509         whether or not the paren't algebra's zero element is a root
 510         of this element's minimal polynomial.
 511
 512         Beware that we can't use the superclass method, because it
 513         relies on the algebra being associative.
 514
 515         SETUP::
 516
 517             sage: from mjo.eja.eja_algebra import random_eja
 518
 519         TESTS:
 520
 521         The identity element is always invertible::
 522
 523             sage: set_random_seed()
 524             sage: J = random_eja()
 525             sage: J.one().is_invertible()
 526             True
 527
 528         The zero element is never invertible in a non-trivial algebra::
 529
 530             sage: set_random_seed()
 531             sage: J = random_eja()
 532             sage: (not J.is_trivial()) and J.zero().is_invertible()
 533             False
 534
 535         """
 536         if self.is_zero():
 537             if self.parent().is_trivial():
 538                 return True
 539             else:
 540                 return False
 541
 542         # In fact, we only need to know if the constant term is non-zero,
 543         # so we can pass in the field's zero element instead.
 544         zero = self.base_ring().zero()
 545         p = self.minimal_polynomial()
 546         return not (p(zero) == zero)
 547
 548
 549     def is_nilpotent(self):
 550         """
 551         Return whether or not some power of this element is zero.
 552
 553         ALGORITHM:
 554
 555         We use Theorem 5 in Chapter III of Koecher, which says that
 556         an element ``x`` is nilpotent if and only if ``x.operator()``
 557         is nilpotent. And it is a basic fact of linear algebra that
 558         an operator on an `n`-dimensional space is nilpotent if and
 559         only if, when raised to the `n`th power, it equals the zero
 560         operator (for example, see Axler Corollary 8.8).
 561
 562         SETUP::
 563
 564             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 565             ....:                                  random_eja)
 566
 567         EXAMPLES::
 568
 569             sage: J = JordanSpinEJA(3)
 570             sage: x = sum(J.gens())
 571             sage: x.is_nilpotent()
 572             False
 573
 574         TESTS:
 575
 576         The identity element is never nilpotent, except in a trivial EJA::
 577
 578             sage: set_random_seed()
 579             sage: J = random_eja()
 580             sage: J.one().is_nilpotent() and not J.is_trivial()
 581             False
 582
 583         The additive identity is always nilpotent::
 584
 585             sage: set_random_seed()
 586             sage: random_eja().zero().is_nilpotent()
 587             True
 588
 589         """
 590         P = self.parent()
 591         zero_operator = P.zero().operator()
 592         return self.operator()**P.dimension() == zero_operator
 593
 594
 595     def is_regular(self):
 596         """
 597         Return whether or not this is a regular element.
 598
 599         SETUP::
 600
 601             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 602             ....:                                  random_eja)
 603
 604         EXAMPLES:
 605
 606         The identity element always has degree one, but any element
 607         linearly-independent from it is regular::
 608
 609             sage: J = JordanSpinEJA(5)
 610             sage: J.one().is_regular()
 611             False
 612             sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
 613             sage: for x in J.gens():
 614             ....:     (J.one() + x).is_regular()
 615             False
 616             True
 617             True
 618             True
 619             True
 620
 621         TESTS:
 622
 623         The zero element should never be regular, unless the parent
 624         algebra has dimension less than or equal to one::
 625
 626             sage: set_random_seed()
 627             sage: J = random_eja()
 628             sage: J.dimension() <= 1 or not J.zero().is_regular()
 629             True
 630
 631         The unit element isn't regular unless the algebra happens to
 632         consist of only its scalar multiples::
 633
 634             sage: set_random_seed()
 635             sage: J = random_eja()
 636             sage: J.dimension() <= 1 or not J.one().is_regular()
 637             True
 638
 639         """
 640         return self.degree() == self.parent().rank()
 641
 642
 643     def degree(self):
 644         """
 645         Return the degree of this element, which is defined to be
 646         the degree of its minimal polynomial.
 647
 648         ALGORITHM:
 649
 650         For now, we skip the messy minimal polynomial computation
 651         and instead return the dimension of the vector space spanned
 652         by the powers of this element. The latter is a bit more
 653         straightforward to compute.
 654
 655         SETUP::
 656
 657             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 658             ....:                                  random_eja)
 659
 660         EXAMPLES::
 661
 662             sage: J = JordanSpinEJA(4)
 663             sage: J.one().degree()
 664             1
 665             sage: e0,e1,e2,e3 = J.gens()
 666             sage: (e0 - e1).degree()
 667             2
 668
 669         In the spin factor algebra (of rank two), all elements that
 670         aren't multiples of the identity are regular::
 671
 672             sage: set_random_seed()
 673             sage: J = JordanSpinEJA.random_instance()
 674             sage: x = J.random_element()
 675             sage: x == x.coefficient(0)*J.one() or x.degree() == 2
 676             True
 677
 678         TESTS:
 679
 680         The zero and unit elements are both of degree one in nontrivial
 681         algebras::
 682
 683             sage: set_random_seed()
 684             sage: J = random_eja()
 685             sage: d = J.zero().degree()
 686             sage: (J.is_trivial() and d == 0) or d == 1
 687             True
 688             sage: d = J.one().degree()
 689             sage: (J.is_trivial() and d == 0) or d == 1
 690             True
 691
 692         Our implementation agrees with the definition::
 693
 694             sage: set_random_seed()
 695             sage: x = random_eja().random_element()
 696             sage: x.degree() == x.minimal_polynomial().degree()
 697             True
 698
 699         """
 700         if self.is_zero() and not self.parent().is_trivial():
 701             # The minimal polynomial of zero in a nontrivial algebra
 702             # is "t"; in a trivial algebra it's "1" by convention
 703             # (it's an empty product).
 704             return 1
 705         return self.subalgebra_generated_by().dimension()
 706
 707
 708     def left_matrix(self):
 709         """
 710         Our parent class defines ``left_matrix`` and ``matrix``
 711         methods whose names are misleading. We don't want them.
 712         """
 713         raise NotImplementedError("use operator().matrix() instead")
 714
 715     matrix = left_matrix
 716
 717
 718     def minimal_polynomial(self):
 719         """
 720         Return the minimal polynomial of this element,
 721         as a function of the variable `t`.
 722
 723         ALGORITHM:
 724
 725         We restrict ourselves to the associative subalgebra
 726         generated by this element, and then return the minimal
 727         polynomial of this element's operator matrix (in that
 728         subalgebra). This works by Baes Proposition 2.3.16.
 729
 730         SETUP::
 731
 732             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 733             ....:                                  RealSymmetricEJA,
 734             ....:                                  random_eja)
 735
 736         TESTS:
 737
 738         The minimal polynomial of the identity and zero elements are
 739         always the same::
 740
 741             sage: set_random_seed()
 742             sage: J = random_eja()
 743             sage: J.one().minimal_polynomial()
 744             t - 1
 745             sage: J.zero().minimal_polynomial()
 746             t
 747
 748         The degree of an element is (by one definition) the degree
 749         of its minimal polynomial::
 750
 751             sage: set_random_seed()
 752             sage: x = random_eja().random_element()
 753             sage: x.degree() == x.minimal_polynomial().degree()
 754             True
 755
 756         The minimal polynomial and the characteristic polynomial coincide
 757         and are known (see Alizadeh, Example 11.11) for all elements of
 758         the spin factor algebra that aren't scalar multiples of the
 759         identity. We require the dimension of the algebra to be at least
 760         two here so that said elements actually exist::
 761
 762             sage: set_random_seed()
 763             sage: n_max = max(2, JordanSpinEJA._max_test_case_size())
 764             sage: n = ZZ.random_element(2, n_max)
 765             sage: J = JordanSpinEJA(n)
 766             sage: y = J.random_element()
 767             sage: while y == y.coefficient(0)*J.one():
 768             ....:     y = J.random_element()
 769             sage: y0 = y.to_vector()[0]
 770             sage: y_bar = y.to_vector()[1:]
 771             sage: actual = y.minimal_polynomial()
 772             sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
 773             sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
 774             sage: bool(actual == expected)
 775             True
 776
 777         The minimal polynomial should always kill its element::
 778
 779             sage: set_random_seed()
 780             sage: x = random_eja().random_element()
 781             sage: p = x.minimal_polynomial()
 782             sage: x.apply_univariate_polynomial(p)
 783             0
 784
 785         The minimal polynomial is invariant under a change of basis,
 786         and in particular, a re-scaling of the basis::
 787
 788             sage: set_random_seed()
 789             sage: n_max = RealSymmetricEJA._max_test_case_size()
 790             sage: n = ZZ.random_element(1, n_max)
 791             sage: J1 = RealSymmetricEJA(n,QQ)
 792             sage: J2 = RealSymmetricEJA(n,QQ,normalize_basis=False)
 793             sage: X = random_matrix(QQ,n)
 794             sage: X = X*X.transpose()
 795             sage: x1 = J1(X)
 796             sage: x2 = J2(X)
 797             sage: x1.minimal_polynomial() == x2.minimal_polynomial()
 798             True
 799
 800         """
 801         if self.is_zero():
 802             # We would generate a zero-dimensional subalgebra
 803             # where the minimal polynomial would be constant.
 804             # That might be correct, but only if *this* algebra
 805             # is trivial too.
 806             if not self.parent().is_trivial():
 807                 # Pretty sure we know what the minimal polynomial of
 808                 # the zero operator is going to be. This ensures
 809                 # consistency of e.g. the polynomial variable returned
 810                 # in the "normal" case without us having to think about it.
 811                 return self.operator().minimal_polynomial()
 812
 813         A = self.subalgebra_generated_by()
 814         return A(self).operator().minimal_polynomial()
 815
 816
 817
 818     def natural_representation(self):
 819         """
 820         Return a more-natural representation of this element.
 821
 822         Every finite-dimensional Euclidean Jordan Algebra is a
 823         direct sum of five simple algebras, four of which comprise
 824         Hermitian matrices. This method returns the original
 825         "natural" representation of this element as a Hermitian
 826         matrix, if it has one. If not, you get the usual representation.
 827
 828         SETUP::
 829
 830             sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
 831             ....:                                  QuaternionHermitianEJA)
 832
 833         EXAMPLES::
 834
 835             sage: J = ComplexHermitianEJA(3)
 836             sage: J.one()
 837             e0 + e3 + e8
 838             sage: J.one().natural_representation()
 839             [1 0 0 0 0 0]
 840             [0 1 0 0 0 0]
 841             [0 0 1 0 0 0]
 842             [0 0 0 1 0 0]
 843             [0 0 0 0 1 0]
 844             [0 0 0 0 0 1]
 845
 846         ::
 847
 848             sage: J = QuaternionHermitianEJA(3)
 849             sage: J.one()
 850             e0 + e5 + e14
 851             sage: J.one().natural_representation()
 852             [1 0 0 0 0 0 0 0 0 0 0 0]
 853             [0 1 0 0 0 0 0 0 0 0 0 0]
 854             [0 0 1 0 0 0 0 0 0 0 0 0]
 855             [0 0 0 1 0 0 0 0 0 0 0 0]
 856             [0 0 0 0 1 0 0 0 0 0 0 0]
 857             [0 0 0 0 0 1 0 0 0 0 0 0]
 858             [0 0 0 0 0 0 1 0 0 0 0 0]
 859             [0 0 0 0 0 0 0 1 0 0 0 0]
 860             [0 0 0 0 0 0 0 0 1 0 0 0]
 861             [0 0 0 0 0 0 0 0 0 1 0 0]
 862             [0 0 0 0 0 0 0 0 0 0 1 0]
 863             [0 0 0 0 0 0 0 0 0 0 0 1]
 864
 865         """
 866         B = self.parent().natural_basis()
 867         W = self.parent().natural_basis_space()
 868         return W.linear_combination(izip(B,self.to_vector()))
 869
 870
 871     def norm(self):
 872         """
 873         The norm of this element with respect to :meth:`inner_product`.
 874
 875         SETUP::
 876
 877             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 878             ....:                                  RealCartesianProductEJA)
 879
 880         EXAMPLES::
 881
 882             sage: J = RealCartesianProductEJA(2)
 883             sage: x = sum(J.gens())
 884             sage: x.norm()
 885             sqrt(2)
 886
 887         ::
 888
 889             sage: J = JordanSpinEJA(4)
 890             sage: x = sum(J.gens())
 891             sage: x.norm()
 892             2
 893
 894         """
 895         return self.inner_product(self).sqrt()
 896
 897
 898     def operator(self):
 899         """
 900         Return the left-multiplication-by-this-element
 901         operator on the ambient algebra.
 902
 903         SETUP::
 904
 905             sage: from mjo.eja.eja_algebra import random_eja
 906
 907         TESTS::
 908
 909             sage: set_random_seed()
 910             sage: J = random_eja()
 911             sage: x,y = J.random_elements(2)
 912             sage: x.operator()(y) == x*y
 913             True
 914             sage: y.operator()(x) == x*y
 915             True
 916
 917         """
 918         P = self.parent()
 919         left_mult_by_self = lambda y: self*y
 920         L = P.module_morphism(function=left_mult_by_self, codomain=P)
 921         return FiniteDimensionalEuclideanJordanAlgebraOperator(
 922                  P,
 923                  P,
 924                  L.matrix() )
 925
 926
 927     def quadratic_representation(self, other=None):
 928         """
 929         Return the quadratic representation of this element.
 930
 931         SETUP::
 932
 933             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 934             ....:                                  random_eja)
 935
 936         EXAMPLES:
 937
 938         The explicit form in the spin factor algebra is given by
 939         Alizadeh's Example 11.12::
 940
 941             sage: set_random_seed()
 942             sage: x = JordanSpinEJA.random_instance().random_element()
 943             sage: x_vec = x.to_vector()
 944             sage: n = x_vec.degree()
 945             sage: x0 = x_vec[0]
 946             sage: x_bar = x_vec[1:]
 947             sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
 948             sage: B = 2*x0*x_bar.row()
 949             sage: C = 2*x0*x_bar.column()
 950             sage: D = matrix.identity(QQ, n-1)
 951             sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
 952             sage: D = D + 2*x_bar.tensor_product(x_bar)
 953             sage: Q = matrix.block(2,2,[A,B,C,D])
 954             sage: Q == x.quadratic_representation().matrix()
 955             True
 956
 957         Test all of the properties from Theorem 11.2 in Alizadeh::
 958
 959             sage: set_random_seed()
 960             sage: J = random_eja()
 961             sage: x,y = J.random_elements(2)
 962             sage: Lx = x.operator()
 963             sage: Lxx = (x*x).operator()
 964             sage: Qx = x.quadratic_representation()
 965             sage: Qy = y.quadratic_representation()
 966             sage: Qxy = x.quadratic_representation(y)
 967             sage: Qex = J.one().quadratic_representation(x)
 968             sage: n = ZZ.random_element(10)
 969             sage: Qxn = (x^n).quadratic_representation()
 970
 971         Property 1:
 972
 973             sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
 974             True
 975
 976         Property 2 (multiply on the right for :trac:`28272`):
 977
 978             sage: alpha = J.base_ring().random_element()
 979             sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
 980             True
 981
 982         Property 3:
 983
 984             sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
 985             True
 986
 987             sage: not x.is_invertible() or (
 988             ....:   ~Qx
 989             ....:   ==
 990             ....:   x.inverse().quadratic_representation() )
 991             True
 992
 993             sage: Qxy(J.one()) == x*y
 994             True
 995
 996         Property 4:
 997
 998             sage: not x.is_invertible() or (
 999             ....:   x.quadratic_representation(x.inverse())*Qx
1000             ....:   == Qx*x.quadratic_representation(x.inverse()) )
1001             True
1002
1003             sage: not x.is_invertible() or (
1004             ....:   x.quadratic_representation(x.inverse())*Qx
1005             ....:   ==
1006             ....:   2*Lx*Qex - Qx )
1007             True
1008
1009             sage: 2*Lx*Qex - Qx == Lxx
1010             True
1011
1012         Property 5:
1013
1014             sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
1015             True
1016
1017         Property 6:
1018
1019             sage: Qxn == (Qx)^n
1020             True
1021
1022         Property 7:
1023
1024             sage: not x.is_invertible() or (
1025             ....:   Qx*x.inverse().operator() == Lx )
1026             True
1027
1028         Property 8:
1029
1030             sage: not x.operator_commutes_with(y) or (
1031             ....:   Qx(y)^n == Qxn(y^n) )
1032             True
1033
1034         """
1035         if other is None:
1036             other=self
1037         elif not other in self.parent():
1038             raise TypeError("'other' must live in the same algebra")
1039
1040         L = self.operator()
1041         M = other.operator()
1042         return ( L*M + M*L - (self*other).operator() )
1043
1044
1045
1046     def spectral_decomposition(self):
1047         """
1048         Return the unique spectral decomposition of this element.
1049
1050         ALGORITHM:
1051
1052         Following Faraut and Korányi's Theorem III.1.1, we restrict this
1053         element's left-multiplication-by operator to the subalgebra it
1054         generates. We then compute the spectral decomposition of that
1055         operator, and the spectral projectors we get back must be the
1056         left-multiplication-by operators for the idempotents we
1057         seek. Thus applying them to the identity element gives us those
1058         idempotents.
1059
1060         Since the eigenvalues are required to be distinct, we take
1061         the spectral decomposition of the zero element to be zero
1062         times the identity element of the algebra (which is idempotent,
1063         obviously).
1064
1065         SETUP::
1066
1067             sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1068
1069         EXAMPLES:
1070
1071         The spectral decomposition of the identity is ``1`` times itself,
1072         and the spectral decomposition of zero is ``0`` times the identity::
1073
1074             sage: J = RealSymmetricEJA(3,AA)
1075             sage: J.one()
1076             e0 + e2 + e5
1077             sage: J.one().spectral_decomposition()
1078             [(1, e0 + e2 + e5)]
1079             sage: J.zero().spectral_decomposition()
1080             [(0, e0 + e2 + e5)]
1081
1082         TESTS::
1083
1084             sage: J = RealSymmetricEJA(4,AA)
1085             sage: x = sum(J.gens())
1086             sage: sd = x.spectral_decomposition()
1087             sage: l0 = sd[0][0]
1088             sage: l1 = sd[1][0]
1089             sage: c0 = sd[0][1]
1090             sage: c1 = sd[1][1]
1091             sage: c0.inner_product(c1) == 0
1092             True
1093             sage: c0.is_idempotent()
1094             True
1095             sage: c1.is_idempotent()
1096             True
1097             sage: c0 + c1 == J.one()
1098             True
1099             sage: l0*c0 + l1*c1 == x
1100             True
1101
1102         """
1103         P = self.parent()
1104         A = self.subalgebra_generated_by(orthonormalize_basis=True)
1105         result = []
1106         for (evalue, proj) in A(self).operator().spectral_decomposition():
1107             result.append( (evalue, proj(A.one()).superalgebra_element()) )
1108         return result
1109
1110     def subalgebra_generated_by(self, orthonormalize_basis=False):
1111         """
1112         Return the associative subalgebra of the parent EJA generated
1113         by this element.
1114
1115         Since our parent algebra is unital, we want "subalgebra" to mean
1116         "unital subalgebra" as well; thus the subalgebra that an element
1117         generates will itself be a Euclidean Jordan algebra after
1118         restricting the algebra operations appropriately. This is the
1119         subalgebra that Faraut and Korányi work with in section II.2, for
1120         example.
1121
1122         SETUP::
1123
1124             sage: from mjo.eja.eja_algebra import random_eja
1125
1126         TESTS:
1127
1128         This subalgebra, being composed of only powers, is associative::
1129
1130             sage: set_random_seed()
1131             sage: x0 = random_eja().random_element()
1132             sage: A = x0.subalgebra_generated_by()
1133             sage: x,y,z = A.random_elements(3)
1134             sage: (x*y)*z == x*(y*z)
1135             True
1136
1137         Squaring in the subalgebra should work the same as in
1138         the superalgebra::
1139
1140             sage: set_random_seed()
1141             sage: x = random_eja().random_element()
1142             sage: A = x.subalgebra_generated_by()
1143             sage: A(x^2) == A(x)*A(x)
1144             True
1145
1146         By definition, the subalgebra generated by the zero element is
1147         the one-dimensional algebra generated by the identity
1148         element... unless the original algebra was trivial, in which
1149         case the subalgebra is trivial too::
1150
1151             sage: set_random_seed()
1152             sage: A = random_eja().zero().subalgebra_generated_by()
1153             sage: (A.is_trivial() and A.dimension() == 0) or A.dimension() == 1
1154             True
1155
1156         """
1157         return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)
1158
1159
1160     def subalgebra_idempotent(self):
1161         """
1162         Find an idempotent in the associative subalgebra I generate
1163         using Proposition 2.3.5 in Baes.
1164
1165         SETUP::
1166
1167             sage: from mjo.eja.eja_algebra import random_eja
1168
1169         TESTS::
1170
1171             sage: set_random_seed()
1172             sage: J = random_eja()
1173             sage: x = J.random_element()
1174             sage: while x.is_nilpotent():
1175             ....:     x = J.random_element()
1176             sage: c = x.subalgebra_idempotent()
1177             sage: c^2 == c
1178             True
1179
1180         """
1181         if self.is_nilpotent():
1182             raise ValueError("this only works with non-nilpotent elements!")
1183
1184         J = self.subalgebra_generated_by()
1185         u = J(self)
1186
1187         # The image of the matrix of left-u^m-multiplication
1188         # will be minimal for some natural number s...
1189         s = 0
1190         minimal_dim = J.dimension()
1191         for i in xrange(1, minimal_dim):
1192             this_dim = (u**i).operator().matrix().image().dimension()
1193             if this_dim < minimal_dim:
1194                 minimal_dim = this_dim
1195                 s = i
1196
1197         # Now minimal_matrix should correspond to the smallest
1198         # non-zero subspace in Baes's (or really, Koecher's)
1199         # proposition.
1200         #
1201         # However, we need to restrict the matrix to work on the
1202         # subspace... or do we? Can't we just solve, knowing that
1203         # A(c) = u^(s+1) should have a solution in the big space,
1204         # too?
1205         #
1206         # Beware, solve_right() means that we're using COLUMN vectors.
1207         # Our FiniteDimensionalAlgebraElement superclass uses rows.
1208         u_next = u**(s+1)
1209         A = u_next.operator().matrix()
1210         c = J.from_vector(A.solve_right(u_next.to_vector()))
1211
1212         # Now c is the idempotent we want, but it still lives in the subalgebra.
1213         return c.superalgebra_element()
1214
1215
1216     def trace(self):
1217         """
1218         Return my trace, the sum of my eigenvalues.
1219
1220         SETUP::
1221
1222             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1223             ....:                                  RealCartesianProductEJA,
1224             ....:                                  random_eja)
1225
1226         EXAMPLES::
1227
1228             sage: J = JordanSpinEJA(3)
1229             sage: x = sum(J.gens())
1230             sage: x.trace()
1231             2
1232
1233         ::
1234
1235             sage: J = RealCartesianProductEJA(5)
1236             sage: J.one().trace()
1237             5
1238
1239         TESTS:
1240
1241         The trace of an element is a real number::
1242
1243             sage: set_random_seed()
1244             sage: J = random_eja()
1245             sage: J.random_element().trace() in RLF
1246             True
1247
1248         """
1249         P = self.parent()
1250         r = P.rank()
1251         p = P._charpoly_coeff(r-1)
1252         # The _charpoly_coeff function already adds the factor of
1253         # -1 to ensure that _charpoly_coeff(r-1) is really what
1254         # appears in front of t^{r-1} in the charpoly. However,
1255         # we want the negative of THAT for the trace.
1256         return -p(*self.to_vector())
1257
1258
1259     def trace_inner_product(self, other):
1260         """
1261         Return the trace inner product of myself and ``other``.
1262
1263         SETUP::
1264
1265             sage: from mjo.eja.eja_algebra import random_eja
1266
1267         TESTS:
1268
1269         The trace inner product is commutative, bilinear, and associative::
1270
1271             sage: set_random_seed()
1272             sage: J = random_eja()
1273             sage: x,y,z = J.random_elements(3)
1274             sage: # commutative
1275             sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1276             True
1277             sage: # bilinear
1278             sage: a = J.base_ring().random_element();
1279             sage: actual = (a*(x+z)).trace_inner_product(y)
1280             sage: expected = ( a*x.trace_inner_product(y) +
1281             ....:              a*z.trace_inner_product(y) )
1282             sage: actual == expected
1283             True
1284             sage: actual = x.trace_inner_product(a*(y+z))
1285             sage: expected = ( a*x.trace_inner_product(y) +
1286             ....:              a*x.trace_inner_product(z) )
1287             sage: actual == expected
1288             True
1289             sage: # associative
1290             sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1291             True
1292
1293         """
1294         if not other in self.parent():
1295             raise TypeError("'other' must live in the same algebra")
1296
1297         return (self*other).trace()
1298
1299
1300     def trace_norm(self):
1301         """
1302         The norm of this element with respect to :meth:`trace_inner_product`.
1303
1304         SETUP::
1305
1306             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1307             ....:                                  RealCartesianProductEJA)
1308
1309         EXAMPLES::
1310
1311             sage: J = RealCartesianProductEJA(2)
1312             sage: x = sum(J.gens())
1313             sage: x.trace_norm()
1314             sqrt(2)
1315
1316         ::
1317
1318             sage: J = JordanSpinEJA(4)
1319             sage: x = sum(J.gens())
1320             sage: x.trace_norm()
1321             2*sqrt(2)
1322
1323         """
1324         return self.trace_inner_product(self).sqrt()