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_element_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_minimal_idempotent(self):
 550         """
 551         Return whether or not this element is a minimal idempotent.
 552
 553
 554         An element of a Euclidean Jordan algebra is a minimal idempotent
 555         if it :meth:`is_idempotent` and if its Peirce subalgebra
 556         corresponding to the eigenvalue ``1`` has dimension ``1`` (Baes,
 557         Proposition 2.7.17).
 558
 559         SETUP::
 560
 561             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 562             ....:                                  RealSymmetricEJA,
 563             ....:                                  random_eja)
 564
 565         WARNING::
 566
 567         This method is sloooooow.
 568
 569         EXAMPLES:
 570
 571         The spectral decomposition of a non-regular element should always
 572         contain at least one non-minimal idempotent::
 573
 574             sage: J = RealSymmetricEJA(3, AA)
 575             sage: x = sum(J.gens())
 576             sage: x.is_regular()
 577             False
 578             sage: [ c.is_minimal_idempotent()
 579             ....:   for (l,c) in x.spectral_decomposition() ]
 580             [False, True]
 581
 582         On the other hand, the spectral decomposition of a regular
 583         element should always be in terms of minimal idempotents::
 584
 585             sage: J = JordanSpinEJA(4, AA)
 586             sage: x = sum( i*J.gens()[i] for i in range(len(J.gens())) )
 587             sage: x.is_regular()
 588             True
 589             sage: [ c.is_minimal_idempotent()
 590             ....:   for (l,c) in x.spectral_decomposition() ]
 591             [True, True]
 592
 593         TESTS:
 594
 595         The identity element is minimal only in an EJA of rank one::
 596
 597             sage: set_random_seed()
 598             sage: J = random_eja()
 599             sage: J.rank() == 1 or not J.one().is_minimal_idempotent()
 600             True
 601
 602         A non-idempotent cannot be a minimal idempotent::
 603
 604             sage: set_random_seed()
 605             sage: J = JordanSpinEJA(4)
 606             sage: x = J.random_element()
 607             sage: (not x.is_idempotent()) and x.is_minimal_idempotent()
 608             False
 609
 610         Proposition 2.7.19 in Baes says that an element is a minimal
 611         idempotent if and only if it's idempotent with trace equal to
 612         unity::
 613
 614             sage: set_random_seed()
 615             sage: J = JordanSpinEJA(4)
 616             sage: x = J.random_element()
 617             sage: expected = (x.is_idempotent() and x.trace() == 1)
 618             sage: actual = x.is_minimal_idempotent()
 619             sage: actual == expected
 620             True
 621
 622         """
 623         if not self.is_idempotent():
 624             return False
 625
 626         (_,_,J1) = self.parent().peirce_decomposition(self)
 627         return (J1.dimension() == 1)
 628
 629
 630     def is_nilpotent(self):
 631         """
 632         Return whether or not some power of this element is zero.
 633
 634         ALGORITHM:
 635
 636         We use Theorem 5 in Chapter III of Koecher, which says that
 637         an element ``x`` is nilpotent if and only if ``x.operator()``
 638         is nilpotent. And it is a basic fact of linear algebra that
 639         an operator on an `n`-dimensional space is nilpotent if and
 640         only if, when raised to the `n`th power, it equals the zero
 641         operator (for example, see Axler Corollary 8.8).
 642
 643         SETUP::
 644
 645             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 646             ....:                                  random_eja)
 647
 648         EXAMPLES::
 649
 650             sage: J = JordanSpinEJA(3)
 651             sage: x = sum(J.gens())
 652             sage: x.is_nilpotent()
 653             False
 654
 655         TESTS:
 656
 657         The identity element is never nilpotent, except in a trivial EJA::
 658
 659             sage: set_random_seed()
 660             sage: J = random_eja()
 661             sage: J.one().is_nilpotent() and not J.is_trivial()
 662             False
 663
 664         The additive identity is always nilpotent::
 665
 666             sage: set_random_seed()
 667             sage: random_eja().zero().is_nilpotent()
 668             True
 669
 670         """
 671         P = self.parent()
 672         zero_operator = P.zero().operator()
 673         return self.operator()**P.dimension() == zero_operator
 674
 675
 676     def is_regular(self):
 677         """
 678         Return whether or not this is a regular element.
 679
 680         SETUP::
 681
 682             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 683             ....:                                  random_eja)
 684
 685         EXAMPLES:
 686
 687         The identity element always has degree one, but any element
 688         linearly-independent from it is regular::
 689
 690             sage: J = JordanSpinEJA(5)
 691             sage: J.one().is_regular()
 692             False
 693             sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
 694             sage: for x in J.gens():
 695             ....:     (J.one() + x).is_regular()
 696             False
 697             True
 698             True
 699             True
 700             True
 701
 702         TESTS:
 703
 704         The zero element should never be regular, unless the parent
 705         algebra has dimension less than or equal to one::
 706
 707             sage: set_random_seed()
 708             sage: J = random_eja()
 709             sage: J.dimension() <= 1 or not J.zero().is_regular()
 710             True
 711
 712         The unit element isn't regular unless the algebra happens to
 713         consist of only its scalar multiples::
 714
 715             sage: set_random_seed()
 716             sage: J = random_eja()
 717             sage: J.dimension() <= 1 or not J.one().is_regular()
 718             True
 719
 720         """
 721         return self.degree() == self.parent().rank()
 722
 723
 724     def degree(self):
 725         """
 726         Return the degree of this element, which is defined to be
 727         the degree of its minimal polynomial.
 728
 729         ALGORITHM:
 730
 731         For now, we skip the messy minimal polynomial computation
 732         and instead return the dimension of the vector space spanned
 733         by the powers of this element. The latter is a bit more
 734         straightforward to compute.
 735
 736         SETUP::
 737
 738             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 739             ....:                                  random_eja)
 740
 741         EXAMPLES::
 742
 743             sage: J = JordanSpinEJA(4)
 744             sage: J.one().degree()
 745             1
 746             sage: e0,e1,e2,e3 = J.gens()
 747             sage: (e0 - e1).degree()
 748             2
 749
 750         In the spin factor algebra (of rank two), all elements that
 751         aren't multiples of the identity are regular::
 752
 753             sage: set_random_seed()
 754             sage: J = JordanSpinEJA.random_instance()
 755             sage: x = J.random_element()
 756             sage: x == x.coefficient(0)*J.one() or x.degree() == 2
 757             True
 758
 759         TESTS:
 760
 761         The zero and unit elements are both of degree one in nontrivial
 762         algebras::
 763
 764             sage: set_random_seed()
 765             sage: J = random_eja()
 766             sage: d = J.zero().degree()
 767             sage: (J.is_trivial() and d == 0) or d == 1
 768             True
 769             sage: d = J.one().degree()
 770             sage: (J.is_trivial() and d == 0) or d == 1
 771             True
 772
 773         Our implementation agrees with the definition::
 774
 775             sage: set_random_seed()
 776             sage: x = random_eja().random_element()
 777             sage: x.degree() == x.minimal_polynomial().degree()
 778             True
 779
 780         """
 781         if self.is_zero() and not self.parent().is_trivial():
 782             # The minimal polynomial of zero in a nontrivial algebra
 783             # is "t"; in a trivial algebra it's "1" by convention
 784             # (it's an empty product).
 785             return 1
 786         return self.subalgebra_generated_by().dimension()
 787
 788
 789     def left_matrix(self):
 790         """
 791         Our parent class defines ``left_matrix`` and ``matrix``
 792         methods whose names are misleading. We don't want them.
 793         """
 794         raise NotImplementedError("use operator().matrix() instead")
 795
 796     matrix = left_matrix
 797
 798
 799     def minimal_polynomial(self):
 800         """
 801         Return the minimal polynomial of this element,
 802         as a function of the variable `t`.
 803
 804         ALGORITHM:
 805
 806         We restrict ourselves to the associative subalgebra
 807         generated by this element, and then return the minimal
 808         polynomial of this element's operator matrix (in that
 809         subalgebra). This works by Baes Proposition 2.3.16.
 810
 811         SETUP::
 812
 813             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 814             ....:                                  RealSymmetricEJA,
 815             ....:                                  TrivialEJA,
 816             ....:                                  random_eja)
 817
 818         EXAMPLES:
 819
 820         Keeping in mind that the polynomial ``1`` evaluates the identity
 821         element (also the zero element) of the trivial algebra, it is clear
 822         that the polynomial ``1`` is the minimal polynomial of the only
 823         element in a trivial algebra::
 824
 825             sage: J = TrivialEJA()
 826             sage: J.one().minimal_polynomial()
 827             1
 828             sage: J.zero().minimal_polynomial()
 829             1
 830
 831         TESTS:
 832
 833         The minimal polynomial of the identity and zero elements are
 834         always the same::
 835
 836             sage: set_random_seed()
 837             sage: J = random_eja(nontrivial=True)
 838             sage: J.one().minimal_polynomial()
 839             t - 1
 840             sage: J.zero().minimal_polynomial()
 841             t
 842
 843         The degree of an element is (by one definition) the degree
 844         of its minimal polynomial::
 845
 846             sage: set_random_seed()
 847             sage: x = random_eja().random_element()
 848             sage: x.degree() == x.minimal_polynomial().degree()
 849             True
 850
 851         The minimal polynomial and the characteristic polynomial coincide
 852         and are known (see Alizadeh, Example 11.11) for all elements of
 853         the spin factor algebra that aren't scalar multiples of the
 854         identity. We require the dimension of the algebra to be at least
 855         two here so that said elements actually exist::
 856
 857             sage: set_random_seed()
 858             sage: n_max = max(2, JordanSpinEJA._max_test_case_size())
 859             sage: n = ZZ.random_element(2, n_max)
 860             sage: J = JordanSpinEJA(n)
 861             sage: y = J.random_element()
 862             sage: while y == y.coefficient(0)*J.one():
 863             ....:     y = J.random_element()
 864             sage: y0 = y.to_vector()[0]
 865             sage: y_bar = y.to_vector()[1:]
 866             sage: actual = y.minimal_polynomial()
 867             sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
 868             sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
 869             sage: bool(actual == expected)
 870             True
 871
 872         The minimal polynomial should always kill its element::
 873
 874             sage: set_random_seed()
 875             sage: x = random_eja().random_element()
 876             sage: p = x.minimal_polynomial()
 877             sage: x.apply_univariate_polynomial(p)
 878             0
 879
 880         The minimal polynomial is invariant under a change of basis,
 881         and in particular, a re-scaling of the basis::
 882
 883             sage: set_random_seed()
 884             sage: n_max = RealSymmetricEJA._max_test_case_size()
 885             sage: n = ZZ.random_element(1, n_max)
 886             sage: J1 = RealSymmetricEJA(n,QQ)
 887             sage: J2 = RealSymmetricEJA(n,QQ,normalize_basis=False)
 888             sage: X = random_matrix(QQ,n)
 889             sage: X = X*X.transpose()
 890             sage: x1 = J1(X)
 891             sage: x2 = J2(X)
 892             sage: x1.minimal_polynomial() == x2.minimal_polynomial()
 893             True
 894
 895         """
 896         if self.is_zero():
 897             # We would generate a zero-dimensional subalgebra
 898             # where the minimal polynomial would be constant.
 899             # That might be correct, but only if *this* algebra
 900             # is trivial too.
 901             if not self.parent().is_trivial():
 902                 # Pretty sure we know what the minimal polynomial of
 903                 # the zero operator is going to be. This ensures
 904                 # consistency of e.g. the polynomial variable returned
 905                 # in the "normal" case without us having to think about it.
 906                 return self.operator().minimal_polynomial()
 907
 908         A = self.subalgebra_generated_by()
 909         return A(self).operator().minimal_polynomial()
 910
 911
 912
 913     def natural_representation(self):
 914         """
 915         Return a more-natural representation of this element.
 916
 917         Every finite-dimensional Euclidean Jordan Algebra is a
 918         direct sum of five simple algebras, four of which comprise
 919         Hermitian matrices. This method returns the original
 920         "natural" representation of this element as a Hermitian
 921         matrix, if it has one. If not, you get the usual representation.
 922
 923         SETUP::
 924
 925             sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
 926             ....:                                  QuaternionHermitianEJA)
 927
 928         EXAMPLES::
 929
 930             sage: J = ComplexHermitianEJA(3)
 931             sage: J.one()
 932             e0 + e3 + e8
 933             sage: J.one().natural_representation()
 934             [1 0 0 0 0 0]
 935             [0 1 0 0 0 0]
 936             [0 0 1 0 0 0]
 937             [0 0 0 1 0 0]
 938             [0 0 0 0 1 0]
 939             [0 0 0 0 0 1]
 940
 941         ::
 942
 943             sage: J = QuaternionHermitianEJA(3)
 944             sage: J.one()
 945             e0 + e5 + e14
 946             sage: J.one().natural_representation()
 947             [1 0 0 0 0 0 0 0 0 0 0 0]
 948             [0 1 0 0 0 0 0 0 0 0 0 0]
 949             [0 0 1 0 0 0 0 0 0 0 0 0]
 950             [0 0 0 1 0 0 0 0 0 0 0 0]
 951             [0 0 0 0 1 0 0 0 0 0 0 0]
 952             [0 0 0 0 0 1 0 0 0 0 0 0]
 953             [0 0 0 0 0 0 1 0 0 0 0 0]
 954             [0 0 0 0 0 0 0 1 0 0 0 0]
 955             [0 0 0 0 0 0 0 0 1 0 0 0]
 956             [0 0 0 0 0 0 0 0 0 1 0 0]
 957             [0 0 0 0 0 0 0 0 0 0 1 0]
 958             [0 0 0 0 0 0 0 0 0 0 0 1]
 959
 960         """
 961         B = self.parent().natural_basis()
 962         W = self.parent().natural_basis_space()
 963         return W.linear_combination(izip(B,self.to_vector()))
 964
 965
 966     def norm(self):
 967         """
 968         The norm of this element with respect to :meth:`inner_product`.
 969
 970         SETUP::
 971
 972             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
 973             ....:                                  RealCartesianProductEJA)
 974
 975         EXAMPLES::
 976
 977             sage: J = RealCartesianProductEJA(2)
 978             sage: x = sum(J.gens())
 979             sage: x.norm()
 980             sqrt(2)
 981
 982         ::
 983
 984             sage: J = JordanSpinEJA(4)
 985             sage: x = sum(J.gens())
 986             sage: x.norm()
 987             2
 988
 989         """
 990         return self.inner_product(self).sqrt()
 991
 992
 993     def operator(self):
 994         """
 995         Return the left-multiplication-by-this-element
 996         operator on the ambient algebra.
 997
 998         SETUP::
 999
1000             sage: from mjo.eja.eja_algebra import random_eja
1001
1002         TESTS::
1003
1004             sage: set_random_seed()
1005             sage: J = random_eja()
1006             sage: x,y = J.random_elements(2)
1007             sage: x.operator()(y) == x*y
1008             True
1009             sage: y.operator()(x) == x*y
1010             True
1011
1012         """
1013         P = self.parent()
1014         left_mult_by_self = lambda y: self*y
1015         L = P.module_morphism(function=left_mult_by_self, codomain=P)
1016         return FiniteDimensionalEuclideanJordanAlgebraOperator(
1017                  P,
1018                  P,
1019                  L.matrix() )
1020
1021
1022     def quadratic_representation(self, other=None):
1023         """
1024         Return the quadratic representation of this element.
1025
1026         SETUP::
1027
1028             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1029             ....:                                  random_eja)
1030
1031         EXAMPLES:
1032
1033         The explicit form in the spin factor algebra is given by
1034         Alizadeh's Example 11.12::
1035
1036             sage: set_random_seed()
1037             sage: x = JordanSpinEJA.random_instance().random_element()
1038             sage: x_vec = x.to_vector()
1039             sage: n = x_vec.degree()
1040             sage: x0 = x_vec[0]
1041             sage: x_bar = x_vec[1:]
1042             sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1043             sage: B = 2*x0*x_bar.row()
1044             sage: C = 2*x0*x_bar.column()
1045             sage: D = matrix.identity(QQ, n-1)
1046             sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1047             sage: D = D + 2*x_bar.tensor_product(x_bar)
1048             sage: Q = matrix.block(2,2,[A,B,C,D])
1049             sage: Q == x.quadratic_representation().matrix()
1050             True
1051
1052         Test all of the properties from Theorem 11.2 in Alizadeh::
1053
1054             sage: set_random_seed()
1055             sage: J = random_eja()
1056             sage: x,y = J.random_elements(2)
1057             sage: Lx = x.operator()
1058             sage: Lxx = (x*x).operator()
1059             sage: Qx = x.quadratic_representation()
1060             sage: Qy = y.quadratic_representation()
1061             sage: Qxy = x.quadratic_representation(y)
1062             sage: Qex = J.one().quadratic_representation(x)
1063             sage: n = ZZ.random_element(10)
1064             sage: Qxn = (x^n).quadratic_representation()
1065
1066         Property 1:
1067
1068             sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1069             True
1070
1071         Property 2 (multiply on the right for :trac:`28272`):
1072
1073             sage: alpha = J.base_ring().random_element()
1074             sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
1075             True
1076
1077         Property 3:
1078
1079             sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
1080             True
1081
1082             sage: not x.is_invertible() or (
1083             ....:   ~Qx
1084             ....:   ==
1085             ....:   x.inverse().quadratic_representation() )
1086             True
1087
1088             sage: Qxy(J.one()) == x*y
1089             True
1090
1091         Property 4:
1092
1093             sage: not x.is_invertible() or (
1094             ....:   x.quadratic_representation(x.inverse())*Qx
1095             ....:   == Qx*x.quadratic_representation(x.inverse()) )
1096             True
1097
1098             sage: not x.is_invertible() or (
1099             ....:   x.quadratic_representation(x.inverse())*Qx
1100             ....:   ==
1101             ....:   2*Lx*Qex - Qx )
1102             True
1103
1104             sage: 2*Lx*Qex - Qx == Lxx
1105             True
1106
1107         Property 5:
1108
1109             sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
1110             True
1111
1112         Property 6:
1113
1114             sage: Qxn == (Qx)^n
1115             True
1116
1117         Property 7:
1118
1119             sage: not x.is_invertible() or (
1120             ....:   Qx*x.inverse().operator() == Lx )
1121             True
1122
1123         Property 8:
1124
1125             sage: not x.operator_commutes_with(y) or (
1126             ....:   Qx(y)^n == Qxn(y^n) )
1127             True
1128
1129         """
1130         if other is None:
1131             other=self
1132         elif not other in self.parent():
1133             raise TypeError("'other' must live in the same algebra")
1134
1135         L = self.operator()
1136         M = other.operator()
1137         return ( L*M + M*L - (self*other).operator() )
1138
1139
1140
1141     def spectral_decomposition(self):
1142         """
1143         Return the unique spectral decomposition of this element.
1144
1145         ALGORITHM:
1146
1147         Following Faraut and Korányi's Theorem III.1.1, we restrict this
1148         element's left-multiplication-by operator to the subalgebra it
1149         generates. We then compute the spectral decomposition of that
1150         operator, and the spectral projectors we get back must be the
1151         left-multiplication-by operators for the idempotents we
1152         seek. Thus applying them to the identity element gives us those
1153         idempotents.
1154
1155         Since the eigenvalues are required to be distinct, we take
1156         the spectral decomposition of the zero element to be zero
1157         times the identity element of the algebra (which is idempotent,
1158         obviously).
1159
1160         SETUP::
1161
1162             sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1163
1164         EXAMPLES:
1165
1166         The spectral decomposition of the identity is ``1`` times itself,
1167         and the spectral decomposition of zero is ``0`` times the identity::
1168
1169             sage: J = RealSymmetricEJA(3,AA)
1170             sage: J.one()
1171             e0 + e2 + e5
1172             sage: J.one().spectral_decomposition()
1173             [(1, e0 + e2 + e5)]
1174             sage: J.zero().spectral_decomposition()
1175             [(0, e0 + e2 + e5)]
1176
1177         TESTS::
1178
1179             sage: J = RealSymmetricEJA(4,AA)
1180             sage: x = sum(J.gens())
1181             sage: sd = x.spectral_decomposition()
1182             sage: l0 = sd[0][0]
1183             sage: l1 = sd[1][0]
1184             sage: c0 = sd[0][1]
1185             sage: c1 = sd[1][1]
1186             sage: c0.inner_product(c1) == 0
1187             True
1188             sage: c0.is_idempotent()
1189             True
1190             sage: c1.is_idempotent()
1191             True
1192             sage: c0 + c1 == J.one()
1193             True
1194             sage: l0*c0 + l1*c1 == x
1195             True
1196
1197         """
1198         P = self.parent()
1199         A = self.subalgebra_generated_by(orthonormalize_basis=True)
1200         result = []
1201         for (evalue, proj) in A(self).operator().spectral_decomposition():
1202             result.append( (evalue, proj(A.one()).superalgebra_element()) )
1203         return result
1204
1205     def subalgebra_generated_by(self, orthonormalize_basis=False):
1206         """
1207         Return the associative subalgebra of the parent EJA generated
1208         by this element.
1209
1210         Since our parent algebra is unital, we want "subalgebra" to mean
1211         "unital subalgebra" as well; thus the subalgebra that an element
1212         generates will itself be a Euclidean Jordan algebra after
1213         restricting the algebra operations appropriately. This is the
1214         subalgebra that Faraut and Korányi work with in section II.2, for
1215         example.
1216
1217         SETUP::
1218
1219             sage: from mjo.eja.eja_algebra import random_eja
1220
1221         TESTS:
1222
1223         This subalgebra, being composed of only powers, is associative::
1224
1225             sage: set_random_seed()
1226             sage: x0 = random_eja().random_element()
1227             sage: A = x0.subalgebra_generated_by()
1228             sage: x,y,z = A.random_elements(3)
1229             sage: (x*y)*z == x*(y*z)
1230             True
1231
1232         Squaring in the subalgebra should work the same as in
1233         the superalgebra::
1234
1235             sage: set_random_seed()
1236             sage: x = random_eja().random_element()
1237             sage: A = x.subalgebra_generated_by()
1238             sage: A(x^2) == A(x)*A(x)
1239             True
1240
1241         By definition, the subalgebra generated by the zero element is
1242         the one-dimensional algebra generated by the identity
1243         element... unless the original algebra was trivial, in which
1244         case the subalgebra is trivial too::
1245
1246             sage: set_random_seed()
1247             sage: A = random_eja().zero().subalgebra_generated_by()
1248             sage: (A.is_trivial() and A.dimension() == 0) or A.dimension() == 1
1249             True
1250
1251         """
1252         return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)
1253
1254
1255     def subalgebra_idempotent(self):
1256         """
1257         Find an idempotent in the associative subalgebra I generate
1258         using Proposition 2.3.5 in Baes.
1259
1260         SETUP::
1261
1262             sage: from mjo.eja.eja_algebra import random_eja
1263
1264         TESTS:
1265
1266         Ensure that we can find an idempotent in a non-trivial algebra
1267         where there are non-nilpotent elements::
1268
1269             sage: set_random_seed()
1270             sage: J = random_eja(nontrivial=True)
1271             sage: x = J.random_element()
1272             sage: while x.is_nilpotent():
1273             ....:     x = J.random_element()
1274             sage: c = x.subalgebra_idempotent()
1275             sage: c^2 == c
1276             True
1277
1278         """
1279         if self.parent().is_trivial():
1280             return self
1281
1282         if self.is_nilpotent():
1283             raise ValueError("this only works with non-nilpotent elements!")
1284
1285         J = self.subalgebra_generated_by()
1286         u = J(self)
1287
1288         # The image of the matrix of left-u^m-multiplication
1289         # will be minimal for some natural number s...
1290         s = 0
1291         minimal_dim = J.dimension()
1292         for i in xrange(1, minimal_dim):
1293             this_dim = (u**i).operator().matrix().image().dimension()
1294             if this_dim < minimal_dim:
1295                 minimal_dim = this_dim
1296                 s = i
1297
1298         # Now minimal_matrix should correspond to the smallest
1299         # non-zero subspace in Baes's (or really, Koecher's)
1300         # proposition.
1301         #
1302         # However, we need to restrict the matrix to work on the
1303         # subspace... or do we? Can't we just solve, knowing that
1304         # A(c) = u^(s+1) should have a solution in the big space,
1305         # too?
1306         #
1307         # Beware, solve_right() means that we're using COLUMN vectors.
1308         # Our FiniteDimensionalAlgebraElement superclass uses rows.
1309         u_next = u**(s+1)
1310         A = u_next.operator().matrix()
1311         c = J.from_vector(A.solve_right(u_next.to_vector()))
1312
1313         # Now c is the idempotent we want, but it still lives in the subalgebra.
1314         return c.superalgebra_element()
1315
1316
1317     def trace(self):
1318         """
1319         Return my trace, the sum of my eigenvalues.
1320
1321         In a trivial algebra, however you want to look at it, the trace is
1322         an empty sum for which we declare the result to be zero.
1323
1324         SETUP::
1325
1326             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1327             ....:                                  RealCartesianProductEJA,
1328             ....:                                  TrivialEJA,
1329             ....:                                  random_eja)
1330
1331         EXAMPLES::
1332
1333             sage: J = TrivialEJA()
1334             sage: J.zero().trace()
1335             0
1336
1337         ::
1338             sage: J = JordanSpinEJA(3)
1339             sage: x = sum(J.gens())
1340             sage: x.trace()
1341             2
1342
1343         ::
1344
1345             sage: J = RealCartesianProductEJA(5)
1346             sage: J.one().trace()
1347             5
1348
1349         TESTS:
1350
1351         The trace of an element is a real number::
1352
1353             sage: set_random_seed()
1354             sage: J = random_eja()
1355             sage: J.random_element().trace() in RLF
1356             True
1357
1358         """
1359         P = self.parent()
1360         r = P.rank()
1361
1362         if r == 0:
1363             # Special case for the trivial algebra where
1364             # the trace is an empty sum.
1365             return P.base_ring().zero()
1366
1367         p = P._charpoly_coeff(r-1)
1368         # The _charpoly_coeff function already adds the factor of
1369         # -1 to ensure that _charpoly_coeff(r-1) is really what
1370         # appears in front of t^{r-1} in the charpoly. However,
1371         # we want the negative of THAT for the trace.
1372         return -p(*self.to_vector())
1373
1374
1375     def trace_inner_product(self, other):
1376         """
1377         Return the trace inner product of myself and ``other``.
1378
1379         SETUP::
1380
1381             sage: from mjo.eja.eja_algebra import random_eja
1382
1383         TESTS:
1384
1385         The trace inner product is commutative, bilinear, and associative::
1386
1387             sage: set_random_seed()
1388             sage: J = random_eja()
1389             sage: x,y,z = J.random_elements(3)
1390             sage: # commutative
1391             sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1392             True
1393             sage: # bilinear
1394             sage: a = J.base_ring().random_element();
1395             sage: actual = (a*(x+z)).trace_inner_product(y)
1396             sage: expected = ( a*x.trace_inner_product(y) +
1397             ....:              a*z.trace_inner_product(y) )
1398             sage: actual == expected
1399             True
1400             sage: actual = x.trace_inner_product(a*(y+z))
1401             sage: expected = ( a*x.trace_inner_product(y) +
1402             ....:              a*x.trace_inner_product(z) )
1403             sage: actual == expected
1404             True
1405             sage: # associative
1406             sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1407             True
1408
1409         """
1410         if not other in self.parent():
1411             raise TypeError("'other' must live in the same algebra")
1412
1413         return (self*other).trace()
1414
1415
1416     def trace_norm(self):
1417         """
1418         The norm of this element with respect to :meth:`trace_inner_product`.
1419
1420         SETUP::
1421
1422             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1423             ....:                                  RealCartesianProductEJA)
1424
1425         EXAMPLES::
1426
1427             sage: J = RealCartesianProductEJA(2)
1428             sage: x = sum(J.gens())
1429             sage: x.trace_norm()
1430             sqrt(2)
1431
1432         ::
1433
1434             sage: J = JordanSpinEJA(4)
1435             sage: x = sum(J.gens())
1436             sage: x.trace_norm()
1437             2*sqrt(2)
1438
1439         """
1440         return self.trace_inner_product(self).sqrt()