2 Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
3 specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
4 are used in optimization, and have some additional nice methods beyond
5 what can be supported in a general Jordan Algebra.
8 from itertools
import repeat
10 from sage
.algebras
.quatalg
.quaternion_algebra
import QuaternionAlgebra
11 from sage
.categories
.magmatic_algebras
import MagmaticAlgebras
12 from sage
.combinat
.free_module
import CombinatorialFreeModule
13 from sage
.matrix
.constructor
import matrix
14 from sage
.matrix
.matrix_space
import MatrixSpace
15 from sage
.misc
.cachefunc
import cached_method
16 from sage
.misc
.lazy_import
import lazy_import
17 from sage
.misc
.prandom
import choice
18 from sage
.misc
.table
import table
19 from sage
.modules
.free_module
import FreeModule
, VectorSpace
20 from sage
.rings
.all
import (ZZ
, QQ
, AA
, QQbar
, RR
, RLF
, CLF
,
23 from mjo
.eja
.eja_element
import FiniteDimensionalEuclideanJordanAlgebraElement
24 lazy_import('mjo.eja.eja_subalgebra',
25 'FiniteDimensionalEuclideanJordanSubalgebra')
26 from mjo
.eja
.eja_utils
import _mat2vec
28 class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule
):
30 def _coerce_map_from_base_ring(self
):
32 Disable the map from the base ring into the algebra.
34 Performing a nonsense conversion like this automatically
35 is counterpedagogical. The fallback is to try the usual
36 element constructor, which should also fail.
40 sage: from mjo.eja.eja_algebra import random_eja
44 sage: set_random_seed()
45 sage: J = random_eja()
47 Traceback (most recent call last):
49 ValueError: not a naturally-represented algebra element
64 sage: from mjo.eja.eja_algebra import (
65 ....: FiniteDimensionalEuclideanJordanAlgebra,
71 By definition, Jordan multiplication commutes::
73 sage: set_random_seed()
74 sage: J = random_eja()
75 sage: x,y = J.random_elements(2)
81 The ``field`` we're given must be real with ``check=True``::
83 sage: JordanSpinEJA(2,QQbar)
84 Traceback (most recent call last):
86 ValueError: field is not real
88 The multiplication table must be square with ``check=True``::
90 sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),()))
91 Traceback (most recent call last):
93 ValueError: multiplication table is not square
97 if not field
.is_subring(RR
):
98 # Note: this does return true for the real algebraic
99 # field, and any quadratic field where we've specified
101 raise ValueError('field is not real')
103 self
._natural
_basis
= natural_basis
106 category
= MagmaticAlgebras(field
).FiniteDimensional()
107 category
= category
.WithBasis().Unital()
109 # The multiplication table had better be square
112 if not all( len(l
) == n
for l
in mult_table
):
113 raise ValueError("multiplication table is not square")
115 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
120 self
.print_options(bracket
='')
122 # The multiplication table we're given is necessarily in terms
123 # of vectors, because we don't have an algebra yet for
124 # anything to be an element of. However, it's faster in the
125 # long run to have the multiplication table be in terms of
126 # algebra elements. We do this after calling the superclass
127 # constructor so that from_vector() knows what to do.
128 self
._multiplication
_table
= [
129 list(map(lambda x
: self
.from_vector(x
), ls
))
134 if not self
._is
_commutative
():
135 raise ValueError("algebra is not commutative")
136 if not self
._is
_jordanian
():
137 raise ValueError("Jordan identity does not hold")
138 if not self
._inner
_product
_is
_associative
():
139 raise ValueError("inner product is not associative")
141 def _element_constructor_(self
, elt
):
143 Construct an element of this algebra from its natural
146 This gets called only after the parent element _call_ method
147 fails to find a coercion for the argument.
151 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
153 ....: RealSymmetricEJA)
157 The identity in `S^n` is converted to the identity in the EJA::
159 sage: J = RealSymmetricEJA(3)
160 sage: I = matrix.identity(QQ,3)
161 sage: J(I) == J.one()
164 This skew-symmetric matrix can't be represented in the EJA::
166 sage: J = RealSymmetricEJA(3)
167 sage: A = matrix(QQ,3, lambda i,j: i-j)
169 Traceback (most recent call last):
171 ArithmeticError: vector is not in free module
175 Ensure that we can convert any element of the two non-matrix
176 simple algebras (whose natural representations are their usual
177 vector representations) back and forth faithfully::
179 sage: set_random_seed()
180 sage: J = HadamardEJA.random_instance()
181 sage: x = J.random_element()
182 sage: J(x.to_vector().column()) == x
184 sage: J = JordanSpinEJA.random_instance()
185 sage: x = J.random_element()
186 sage: J(x.to_vector().column()) == x
190 msg
= "not a naturally-represented algebra element"
192 # The superclass implementation of random_element()
193 # needs to be able to coerce "0" into the algebra.
195 elif elt
in self
.base_ring():
196 # Ensure that no base ring -> algebra coercion is performed
197 # by this method. There's some stupidity in sage that would
198 # otherwise propagate to this method; for example, sage thinks
199 # that the integer 3 belongs to the space of 2-by-2 matrices.
200 raise ValueError(msg
)
202 natural_basis
= self
.natural_basis()
203 basis_space
= natural_basis
[0].matrix_space()
204 if elt
not in basis_space
:
205 raise ValueError(msg
)
207 # Thanks for nothing! Matrix spaces aren't vector spaces in
208 # Sage, so we have to figure out its natural-basis coordinates
209 # ourselves. We use the basis space's ring instead of the
210 # element's ring because the basis space might be an algebraic
211 # closure whereas the base ring of the 3-by-3 identity matrix
212 # could be QQ instead of QQbar.
213 V
= VectorSpace(basis_space
.base_ring(), elt
.nrows()*elt
.ncols())
214 W
= V
.span_of_basis( _mat2vec(s
) for s
in natural_basis
)
215 coords
= W
.coordinate_vector(_mat2vec(elt
))
216 return self
.from_vector(coords
)
219 def _max_test_case_size():
221 Return an integer "size" that is an upper bound on the size of
222 this algebra when it is used in a random test
223 case. Unfortunately, the term "size" is quite vague -- when
224 dealing with `R^n` under either the Hadamard or Jordan spin
225 product, the "size" refers to the dimension `n`. When dealing
226 with a matrix algebra (real symmetric or complex/quaternion
227 Hermitian), it refers to the size of the matrix, which is
228 far less than the dimension of the underlying vector space.
230 We default to five in this class, which is safe in `R^n`. The
231 matrix algebra subclasses (or any class where the "size" is
232 interpreted to be far less than the dimension) should override
233 with a smaller number.
239 Return a string representation of ``self``.
243 sage: from mjo.eja.eja_algebra import JordanSpinEJA
247 Ensure that it says what we think it says::
249 sage: JordanSpinEJA(2, field=AA)
250 Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
251 sage: JordanSpinEJA(3, field=RDF)
252 Euclidean Jordan algebra of dimension 3 over Real Double Field
255 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
256 return fmt
.format(self
.dimension(), self
.base_ring())
258 def product_on_basis(self
, i
, j
):
259 return self
._multiplication
_table
[i
][j
]
261 def _is_commutative(self
):
263 Whether or not this algebra's multiplication table is commutative.
265 This method should of course always return ``True``, unless
266 this algebra was constructed with ``check=False`` and passed
267 an invalid multiplication table.
269 return all( self
.product_on_basis(i
,j
) == self
.product_on_basis(i
,j
)
270 for i
in range(self
.dimension())
271 for j
in range(self
.dimension()) )
273 def _is_jordanian(self
):
275 Whether or not this algebra's multiplication table respects the
276 Jordan identity `(x^{2})(xy) = x(x^{2}y)`.
278 We only check one arrangement of `x` and `y`, so for a
279 ``True`` result to be truly true, you should also check
280 :meth:`_is_commutative`. This method should of course always
281 return ``True``, unless this algebra was constructed with
282 ``check=False`` and passed an invalid multiplication table.
284 return all( (self
.monomial(i
)**2)*(self
.monomial(i
)*self
.monomial(j
))
286 (self
.monomial(i
))*((self
.monomial(i
)**2)*self
.monomial(j
))
287 for i
in range(self
.dimension())
288 for j
in range(self
.dimension()) )
290 def _inner_product_is_associative(self
):
292 Return whether or not this algebra's inner product `B` is
293 associative; that is, whether or not `B(xy,z) = B(x,yz)`.
295 This method should of course always return ``True``, unless
296 this algebra was constructed with ``check=False`` and passed
297 an invalid multiplication table.
300 # Used to check whether or not something is zero in an inexact
301 # ring. This number is sufficient to allow the construction of
302 # QuaternionHermitianEJA(2, RDF) with check=True.
305 for i
in range(self
.dimension()):
306 for j
in range(self
.dimension()):
307 for k
in range(self
.dimension()):
311 diff
= (x
*y
).inner_product(z
) - x
.inner_product(y
*z
)
313 if self
.base_ring().is_exact():
317 if diff
.abs() > epsilon
:
323 def characteristic_polynomial_of(self
):
325 Return the algebra's "characteristic polynomial of" function,
326 which is itself a multivariate polynomial that, when evaluated
327 at the coordinates of some algebra element, returns that
328 element's characteristic polynomial.
330 The resulting polynomial has `n+1` variables, where `n` is the
331 dimension of this algebra. The first `n` variables correspond to
332 the coordinates of an algebra element: when evaluated at the
333 coordinates of an algebra element with respect to a certain
334 basis, the result is a univariate polynomial (in the one
335 remaining variable ``t``), namely the characteristic polynomial
340 sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA
344 The characteristic polynomial in the spin algebra is given in
345 Alizadeh, Example 11.11::
347 sage: J = JordanSpinEJA(3)
348 sage: p = J.characteristic_polynomial_of(); p
349 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
350 sage: xvec = J.one().to_vector()
354 By definition, the characteristic polynomial is a monic
355 degree-zero polynomial in a rank-zero algebra. Note that
356 Cayley-Hamilton is indeed satisfied since the polynomial
357 ``1`` evaluates to the identity element of the algebra on
360 sage: J = TrivialEJA()
361 sage: J.characteristic_polynomial_of()
368 # The list of coefficient polynomials a_0, a_1, a_2, ..., a_(r-1).
369 a
= self
._charpoly
_coefficients
()
371 # We go to a bit of trouble here to reorder the
372 # indeterminates, so that it's easier to evaluate the
373 # characteristic polynomial at x's coordinates and get back
374 # something in terms of t, which is what we want.
375 S
= PolynomialRing(self
.base_ring(),'t')
379 S
= PolynomialRing(S
, R
.variable_names())
382 return (t
**r
+ sum( a
[k
]*(t
**k
) for k
in range(r
) ))
385 def inner_product(self
, x
, y
):
387 The inner product associated with this Euclidean Jordan algebra.
389 Defaults to the trace inner product, but can be overridden by
390 subclasses if they are sure that the necessary properties are
395 sage: from mjo.eja.eja_algebra import random_eja
399 Our inner product is "associative," which means the following for
400 a symmetric bilinear form::
402 sage: set_random_seed()
403 sage: J = random_eja()
404 sage: x,y,z = J.random_elements(3)
405 sage: (x*y).inner_product(z) == y.inner_product(x*z)
409 X
= x
.natural_representation()
410 Y
= y
.natural_representation()
411 return self
.natural_inner_product(X
,Y
)
414 def is_trivial(self
):
416 Return whether or not this algebra is trivial.
418 A trivial algebra contains only the zero element.
422 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
427 sage: J = ComplexHermitianEJA(3)
433 sage: J = TrivialEJA()
438 return self
.dimension() == 0
441 def multiplication_table(self
):
443 Return a visual representation of this algebra's multiplication
444 table (on basis elements).
448 sage: from mjo.eja.eja_algebra import JordanSpinEJA
452 sage: J = JordanSpinEJA(4)
453 sage: J.multiplication_table()
454 +----++----+----+----+----+
455 | * || e0 | e1 | e2 | e3 |
456 +====++====+====+====+====+
457 | e0 || e0 | e1 | e2 | e3 |
458 +----++----+----+----+----+
459 | e1 || e1 | e0 | 0 | 0 |
460 +----++----+----+----+----+
461 | e2 || e2 | 0 | e0 | 0 |
462 +----++----+----+----+----+
463 | e3 || e3 | 0 | 0 | e0 |
464 +----++----+----+----+----+
467 M
= list(self
._multiplication
_table
) # copy
468 for i
in range(len(M
)):
469 # M had better be "square"
470 M
[i
] = [self
.monomial(i
)] + M
[i
]
471 M
= [["*"] + list(self
.gens())] + M
472 return table(M
, header_row
=True, header_column
=True, frame
=True)
475 def natural_basis(self
):
477 Return a more-natural representation of this algebra's basis.
479 Every finite-dimensional Euclidean Jordan Algebra is a direct
480 sum of five simple algebras, four of which comprise Hermitian
481 matrices. This method returns the original "natural" basis
482 for our underlying vector space. (Typically, the natural basis
483 is used to construct the multiplication table in the first place.)
485 Note that this will always return a matrix. The standard basis
486 in `R^n` will be returned as `n`-by-`1` column matrices.
490 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
491 ....: RealSymmetricEJA)
495 sage: J = RealSymmetricEJA(2)
497 Finite family {0: e0, 1: e1, 2: e2}
498 sage: J.natural_basis()
500 [1 0] [ 0 0.7071067811865475?] [0 0]
501 [0 0], [0.7071067811865475? 0], [0 1]
506 sage: J = JordanSpinEJA(2)
508 Finite family {0: e0, 1: e1}
509 sage: J.natural_basis()
516 if self
._natural
_basis
is None:
517 M
= self
.natural_basis_space()
518 return tuple( M(b
.to_vector()) for b
in self
.basis() )
520 return self
._natural
_basis
523 def natural_basis_space(self
):
525 Return the matrix space in which this algebra's natural basis
528 Generally this will be an `n`-by-`1` column-vector space,
529 except when the algebra is trivial. There it's `n`-by-`n`
530 (where `n` is zero), to ensure that two elements of the
531 natural basis space (empty matrices) can be multiplied.
533 if self
.is_trivial():
534 return MatrixSpace(self
.base_ring(), 0)
535 elif self
._natural
_basis
is None or len(self
._natural
_basis
) == 0:
536 return MatrixSpace(self
.base_ring(), self
.dimension(), 1)
538 return self
._natural
_basis
[0].matrix_space()
542 def natural_inner_product(X
,Y
):
544 Compute the inner product of two naturally-represented elements.
546 For example in the real symmetric matrix EJA, this will compute
547 the trace inner-product of two n-by-n symmetric matrices. The
548 default should work for the real cartesian product EJA, the
549 Jordan spin EJA, and the real symmetric matrices. The others
550 will have to be overridden.
552 return (X
.conjugate_transpose()*Y
).trace()
558 Return the unit element of this algebra.
562 sage: from mjo.eja.eja_algebra import (HadamardEJA,
567 sage: J = HadamardEJA(5)
569 e0 + e1 + e2 + e3 + e4
573 The identity element acts like the identity::
575 sage: set_random_seed()
576 sage: J = random_eja()
577 sage: x = J.random_element()
578 sage: J.one()*x == x and x*J.one() == x
581 The matrix of the unit element's operator is the identity::
583 sage: set_random_seed()
584 sage: J = random_eja()
585 sage: actual = J.one().operator().matrix()
586 sage: expected = matrix.identity(J.base_ring(), J.dimension())
587 sage: actual == expected
591 # We can brute-force compute the matrices of the operators
592 # that correspond to the basis elements of this algebra.
593 # If some linear combination of those basis elements is the
594 # algebra identity, then the same linear combination of
595 # their matrices has to be the identity matrix.
597 # Of course, matrices aren't vectors in sage, so we have to
598 # appeal to the "long vectors" isometry.
599 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
601 # Now we use basis linear algebra to find the coefficients,
602 # of the matrices-as-vectors-linear-combination, which should
603 # work for the original algebra basis too.
604 A
= matrix
.column(self
.base_ring(), oper_vecs
)
606 # We used the isometry on the left-hand side already, but we
607 # still need to do it for the right-hand side. Recall that we
608 # wanted something that summed to the identity matrix.
609 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
611 # Now if there's an identity element in the algebra, this should work.
612 coeffs
= A
.solve_right(b
)
613 return self
.linear_combination(zip(self
.gens(), coeffs
))
616 def peirce_decomposition(self
, c
):
618 The Peirce decomposition of this algebra relative to the
621 In the future, this can be extended to a complete system of
622 orthogonal idempotents.
626 - ``c`` -- an idempotent of this algebra.
630 A triple (J0, J5, J1) containing two subalgebras and one subspace
633 - ``J0`` -- the algebra on the eigenspace of ``c.operator()``
634 corresponding to the eigenvalue zero.
636 - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()``
637 corresponding to the eigenvalue one-half.
639 - ``J1`` -- the algebra on the eigenspace of ``c.operator()``
640 corresponding to the eigenvalue one.
642 These are the only possible eigenspaces for that operator, and this
643 algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are
644 orthogonal, and are subalgebras of this algebra with the appropriate
649 sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA
653 The canonical example comes from the symmetric matrices, which
654 decompose into diagonal and off-diagonal parts::
656 sage: J = RealSymmetricEJA(3)
657 sage: C = matrix(QQ, [ [1,0,0],
661 sage: J0,J5,J1 = J.peirce_decomposition(c)
663 Euclidean Jordan algebra of dimension 1...
665 Vector space of degree 6 and dimension 2...
667 Euclidean Jordan algebra of dimension 3...
668 sage: J0.one().natural_representation()
672 sage: orig_df = AA.options.display_format
673 sage: AA.options.display_format = 'radical'
674 sage: J.from_vector(J5.basis()[0]).natural_representation()
678 sage: J.from_vector(J5.basis()[1]).natural_representation()
682 sage: AA.options.display_format = orig_df
683 sage: J1.one().natural_representation()
690 Every algebra decomposes trivially with respect to its identity
693 sage: set_random_seed()
694 sage: J = random_eja()
695 sage: J0,J5,J1 = J.peirce_decomposition(J.one())
696 sage: J0.dimension() == 0 and J5.dimension() == 0
698 sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
701 The decomposition is into eigenspaces, and its components are
702 therefore necessarily orthogonal. Moreover, the identity
703 elements in the two subalgebras are the projections onto their
704 respective subspaces of the superalgebra's identity element::
706 sage: set_random_seed()
707 sage: J = random_eja()
708 sage: x = J.random_element()
709 sage: if not J.is_trivial():
710 ....: while x.is_nilpotent():
711 ....: x = J.random_element()
712 sage: c = x.subalgebra_idempotent()
713 sage: J0,J5,J1 = J.peirce_decomposition(c)
715 sage: for (w,y,z) in zip(J0.basis(), J5.basis(), J1.basis()):
716 ....: w = w.superalgebra_element()
717 ....: y = J.from_vector(y)
718 ....: z = z.superalgebra_element()
719 ....: ipsum += w.inner_product(y).abs()
720 ....: ipsum += w.inner_product(z).abs()
721 ....: ipsum += y.inner_product(z).abs()
724 sage: J1(c) == J1.one()
726 sage: J0(J.one() - c) == J0.one()
730 if not c
.is_idempotent():
731 raise ValueError("element is not idempotent: %s" % c
)
733 # Default these to what they should be if they turn out to be
734 # trivial, because eigenspaces_left() won't return eigenvalues
735 # corresponding to trivial spaces (e.g. it returns only the
736 # eigenspace corresponding to lambda=1 if you take the
737 # decomposition relative to the identity element).
738 trivial
= FiniteDimensionalEuclideanJordanSubalgebra(self
, ())
739 J0
= trivial
# eigenvalue zero
740 J5
= VectorSpace(self
.base_ring(), 0) # eigenvalue one-half
741 J1
= trivial
# eigenvalue one
743 for (eigval
, eigspace
) in c
.operator().matrix().right_eigenspaces():
744 if eigval
== ~
(self
.base_ring()(2)):
747 gens
= tuple( self
.from_vector(b
) for b
in eigspace
.basis() )
748 subalg
= FiniteDimensionalEuclideanJordanSubalgebra(self
, gens
)
754 raise ValueError("unexpected eigenvalue: %s" % eigval
)
759 def random_element(self
, thorough
=False):
761 Return a random element of this algebra.
763 Our algebra superclass method only returns a linear
764 combination of at most two basis elements. We instead
765 want the vector space "random element" method that
766 returns a more diverse selection.
770 - ``thorough`` -- (boolean; default False) whether or not we
771 should generate irrational coefficients for the random
772 element when our base ring is irrational; this slows the
773 algebra operations to a crawl, but any truly random method
777 # For a general base ring... maybe we can trust this to do the
778 # right thing? Unlikely, but.
779 V
= self
.vector_space()
780 v
= V
.random_element()
782 if self
.base_ring() is AA
:
783 # The "random element" method of the algebraic reals is
784 # stupid at the moment, and only returns integers between
785 # -2 and 2, inclusive:
787 # https://trac.sagemath.org/ticket/30875
789 # Instead, we implement our own "random vector" method,
790 # and then coerce that into the algebra. We use the vector
791 # space degree here instead of the dimension because a
792 # subalgebra could (for example) be spanned by only two
793 # vectors, each with five coordinates. We need to
794 # generate all five coordinates.
796 v
*= QQbar
.random_element().real()
798 v
*= QQ
.random_element()
800 return self
.from_vector(V
.coordinate_vector(v
))
802 def random_elements(self
, count
, thorough
=False):
804 Return ``count`` random elements as a tuple.
808 - ``thorough`` -- (boolean; default False) whether or not we
809 should generate irrational coefficients for the random
810 elements when our base ring is irrational; this slows the
811 algebra operations to a crawl, but any truly random method
816 sage: from mjo.eja.eja_algebra import JordanSpinEJA
820 sage: J = JordanSpinEJA(3)
821 sage: x,y,z = J.random_elements(3)
822 sage: all( [ x in J, y in J, z in J ])
824 sage: len( J.random_elements(10) ) == 10
828 return tuple( self
.random_element(thorough
)
829 for idx
in range(count
) )
832 def random_instance(cls
, field
=AA
, **kwargs
):
834 Return a random instance of this type of algebra.
836 Beware, this will crash for "most instances" because the
837 constructor below looks wrong.
839 if cls
is TrivialEJA
:
840 # The TrivialEJA class doesn't take an "n" argument because
844 n
= ZZ
.random_element(cls
._max
_test
_case
_size
() + 1)
845 return cls(n
, field
, **kwargs
)
848 def _charpoly_coefficients(self
):
850 The `r` polynomial coefficients of the "characteristic polynomial
854 var_names
= [ "X" + str(z
) for z
in range(1,n
+1) ]
855 R
= PolynomialRing(self
.base_ring(), var_names
)
857 F
= R
.fraction_field()
860 # From a result in my book, these are the entries of the
861 # basis representation of L_x.
862 return sum( vars[k
]*self
.monomial(k
).operator().matrix()[i
,j
]
865 L_x
= matrix(F
, n
, n
, L_x_i_j
)
868 if self
.rank
.is_in_cache():
870 # There's no need to pad the system with redundant
871 # columns if we *know* they'll be redundant.
874 # Compute an extra power in case the rank is equal to
875 # the dimension (otherwise, we would stop at x^(r-1)).
876 x_powers
= [ (L_x
**k
)*self
.one().to_vector()
877 for k
in range(n
+1) ]
878 A
= matrix
.column(F
, x_powers
[:n
])
879 AE
= A
.extended_echelon_form()
886 # The theory says that only the first "r" coefficients are
887 # nonzero, and they actually live in the original polynomial
888 # ring and not the fraction field. We negate them because
889 # in the actual characteristic polynomial, they get moved
890 # to the other side where x^r lives.
891 return -A_rref
.solve_right(E
*b
).change_ring(R
)[:r
]
896 Return the rank of this EJA.
898 This is a cached method because we know the rank a priori for
899 all of the algebras we can construct. Thus we can avoid the
900 expensive ``_charpoly_coefficients()`` call unless we truly
901 need to compute the whole characteristic polynomial.
905 sage: from mjo.eja.eja_algebra import (HadamardEJA,
907 ....: RealSymmetricEJA,
908 ....: ComplexHermitianEJA,
909 ....: QuaternionHermitianEJA,
914 The rank of the Jordan spin algebra is always two::
916 sage: JordanSpinEJA(2).rank()
918 sage: JordanSpinEJA(3).rank()
920 sage: JordanSpinEJA(4).rank()
923 The rank of the `n`-by-`n` Hermitian real, complex, or
924 quaternion matrices is `n`::
926 sage: RealSymmetricEJA(4).rank()
928 sage: ComplexHermitianEJA(3).rank()
930 sage: QuaternionHermitianEJA(2).rank()
935 Ensure that every EJA that we know how to construct has a
936 positive integer rank, unless the algebra is trivial in
937 which case its rank will be zero::
939 sage: set_random_seed()
940 sage: J = random_eja()
944 sage: r > 0 or (r == 0 and J.is_trivial())
947 Ensure that computing the rank actually works, since the ranks
948 of all simple algebras are known and will be cached by default::
950 sage: J = HadamardEJA(4)
951 sage: J.rank.clear_cache()
957 sage: J = JordanSpinEJA(4)
958 sage: J.rank.clear_cache()
964 sage: J = RealSymmetricEJA(3)
965 sage: J.rank.clear_cache()
971 sage: J = ComplexHermitianEJA(2)
972 sage: J.rank.clear_cache()
978 sage: J = QuaternionHermitianEJA(2)
979 sage: J.rank.clear_cache()
983 return len(self
._charpoly
_coefficients
())
986 def vector_space(self
):
988 Return the vector space that underlies this algebra.
992 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
996 sage: J = RealSymmetricEJA(2)
997 sage: J.vector_space()
998 Vector space of dimension 3 over...
1001 return self
.zero().to_vector().parent().ambient_vector_space()
1004 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
1007 class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1009 Return the Euclidean Jordan Algebra corresponding to the set
1010 `R^n` under the Hadamard product.
1012 Note: this is nothing more than the Cartesian product of ``n``
1013 copies of the spin algebra. Once Cartesian product algebras
1014 are implemented, this can go.
1018 sage: from mjo.eja.eja_algebra import HadamardEJA
1022 This multiplication table can be verified by hand::
1024 sage: J = HadamardEJA(3)
1025 sage: e0,e1,e2 = J.gens()
1041 We can change the generator prefix::
1043 sage: HadamardEJA(3, prefix='r').gens()
1047 def __init__(self
, n
, field
=AA
, **kwargs
):
1048 V
= VectorSpace(field
, n
)
1049 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in range(n
) ]
1052 fdeja
= super(HadamardEJA
, self
)
1053 fdeja
.__init
__(field
, mult_table
, **kwargs
)
1054 self
.rank
.set_cache(n
)
1056 def inner_product(self
, x
, y
):
1058 Faster to reimplement than to use natural representations.
1062 sage: from mjo.eja.eja_algebra import HadamardEJA
1066 Ensure that this is the usual inner product for the algebras
1069 sage: set_random_seed()
1070 sage: J = HadamardEJA.random_instance()
1071 sage: x,y = J.random_elements(2)
1072 sage: X = x.natural_representation()
1073 sage: Y = y.natural_representation()
1074 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
1078 return x
.to_vector().inner_product(y
.to_vector())
1081 def random_eja(field
=AA
):
1083 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1087 sage: from mjo.eja.eja_algebra import random_eja
1092 Euclidean Jordan algebra of dimension...
1095 classname
= choice([TrivialEJA
,
1099 ComplexHermitianEJA
,
1100 QuaternionHermitianEJA
])
1101 return classname
.random_instance(field
=field
)
1106 class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
1108 def _max_test_case_size():
1109 # Play it safe, since this will be squared and the underlying
1110 # field can have dimension 4 (quaternions) too.
1113 def __init__(self
, field
, basis
, normalize_basis
=True, **kwargs
):
1115 Compared to the superclass constructor, we take a basis instead of
1116 a multiplication table because the latter can be computed in terms
1117 of the former when the product is known (like it is here).
1119 # Used in this class's fast _charpoly_coefficients() override.
1120 self
._basis
_normalizers
= None
1122 # We're going to loop through this a few times, so now's a good
1123 # time to ensure that it isn't a generator expression.
1124 basis
= tuple(basis
)
1126 if len(basis
) > 1 and normalize_basis
:
1127 # We'll need sqrt(2) to normalize the basis, and this
1128 # winds up in the multiplication table, so the whole
1129 # algebra needs to be over the field extension.
1130 R
= PolynomialRing(field
, 'z')
1133 if p
.is_irreducible():
1134 field
= field
.extension(p
, 'sqrt2', embedding
=RLF(2).sqrt())
1135 basis
= tuple( s
.change_ring(field
) for s
in basis
)
1136 self
._basis
_normalizers
= tuple(
1137 ~
(self
.natural_inner_product(s
,s
).sqrt()) for s
in basis
)
1138 basis
= tuple(s
*c
for (s
,c
) in zip(basis
,self
._basis
_normalizers
))
1140 Qs
= self
.multiplication_table_from_matrix_basis(basis
)
1142 fdeja
= super(MatrixEuclideanJordanAlgebra
, self
)
1143 fdeja
.__init
__(field
, Qs
, natural_basis
=basis
, **kwargs
)
1148 def _charpoly_coefficients(self
):
1150 Override the parent method with something that tries to compute
1151 over a faster (non-extension) field.
1153 if self
._basis
_normalizers
is None:
1154 # We didn't normalize, so assume that the basis we started
1155 # with had entries in a nice field.
1156 return super(MatrixEuclideanJordanAlgebra
, self
)._charpoly
_coefficients
()
1158 basis
= ( (b
/n
) for (b
,n
) in zip(self
.natural_basis(),
1159 self
._basis
_normalizers
) )
1161 # Do this over the rationals and convert back at the end.
1162 # Only works because we know the entries of the basis are
1164 J
= MatrixEuclideanJordanAlgebra(QQ
,
1166 normalize_basis
=False)
1167 a
= J
._charpoly
_coefficients
()
1169 # Unfortunately, changing the basis does change the
1170 # coefficients of the characteristic polynomial, but since
1171 # these are really the coefficients of the "characteristic
1172 # polynomial of" function, everything is still nice and
1173 # unevaluated. It's therefore "obvious" how scaling the
1174 # basis affects the coordinate variables X1, X2, et
1175 # cetera. Scaling the first basis vector up by "n" adds a
1176 # factor of 1/n into every "X1" term, for example. So here
1177 # we simply undo the basis_normalizer scaling that we
1178 # performed earlier.
1180 # The a[0] access here is safe because trivial algebras
1181 # won't have any basis normalizers and therefore won't
1182 # make it to this "else" branch.
1183 XS
= a
[0].parent().gens()
1184 subs_dict
= { XS
[i
]: self
._basis
_normalizers
[i
]*XS
[i
]
1185 for i
in range(len(XS
)) }
1186 return tuple( a_i
.subs(subs_dict
) for a_i
in a
)
1190 def multiplication_table_from_matrix_basis(basis
):
1192 At least three of the five simple Euclidean Jordan algebras have the
1193 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1194 multiplication on the right is matrix multiplication. Given a basis
1195 for the underlying matrix space, this function returns a
1196 multiplication table (obtained by looping through the basis
1197 elements) for an algebra of those matrices.
1199 # In S^2, for example, we nominally have four coordinates even
1200 # though the space is of dimension three only. The vector space V
1201 # is supposed to hold the entire long vector, and the subspace W
1202 # of V will be spanned by the vectors that arise from symmetric
1203 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1207 field
= basis
[0].base_ring()
1208 dimension
= basis
[0].nrows()
1210 V
= VectorSpace(field
, dimension
**2)
1211 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
1213 mult_table
= [[W
.zero() for j
in range(n
)] for i
in range(n
)]
1216 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
1217 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
1225 Embed the matrix ``M`` into a space of real matrices.
1227 The matrix ``M`` can have entries in any field at the moment:
1228 the real numbers, complex numbers, or quaternions. And although
1229 they are not a field, we can probably support octonions at some
1230 point, too. This function returns a real matrix that "acts like"
1231 the original with respect to matrix multiplication; i.e.
1233 real_embed(M*N) = real_embed(M)*real_embed(N)
1236 raise NotImplementedError
1240 def real_unembed(M
):
1242 The inverse of :meth:`real_embed`.
1244 raise NotImplementedError
1248 def natural_inner_product(cls
,X
,Y
):
1249 Xu
= cls
.real_unembed(X
)
1250 Yu
= cls
.real_unembed(Y
)
1251 tr
= (Xu
*Yu
).trace()
1254 # It's real already.
1257 # Otherwise, try the thing that works for complex numbers; and
1258 # if that doesn't work, the thing that works for quaternions.
1260 return tr
.vector()[0] # real part, imag part is index 1
1261 except AttributeError:
1262 # A quaternions doesn't have a vector() method, but does
1263 # have coefficient_tuple() method that returns the
1264 # coefficients of 1, i, j, and k -- in that order.
1265 return tr
.coefficient_tuple()[0]
1268 class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1272 The identity function, for embedding real matrices into real
1278 def real_unembed(M
):
1280 The identity function, for unembedding real matrices from real
1286 class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra
):
1288 The rank-n simple EJA consisting of real symmetric n-by-n
1289 matrices, the usual symmetric Jordan product, and the trace inner
1290 product. It has dimension `(n^2 + n)/2` over the reals.
1294 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1298 sage: J = RealSymmetricEJA(2)
1299 sage: e0, e1, e2 = J.gens()
1307 In theory, our "field" can be any subfield of the reals::
1309 sage: RealSymmetricEJA(2, RDF)
1310 Euclidean Jordan algebra of dimension 3 over Real Double Field
1311 sage: RealSymmetricEJA(2, RR)
1312 Euclidean Jordan algebra of dimension 3 over Real Field with
1313 53 bits of precision
1317 The dimension of this algebra is `(n^2 + n) / 2`::
1319 sage: set_random_seed()
1320 sage: n_max = RealSymmetricEJA._max_test_case_size()
1321 sage: n = ZZ.random_element(1, n_max)
1322 sage: J = RealSymmetricEJA(n)
1323 sage: J.dimension() == (n^2 + n)/2
1326 The Jordan multiplication is what we think it is::
1328 sage: set_random_seed()
1329 sage: J = RealSymmetricEJA.random_instance()
1330 sage: x,y = J.random_elements(2)
1331 sage: actual = (x*y).natural_representation()
1332 sage: X = x.natural_representation()
1333 sage: Y = y.natural_representation()
1334 sage: expected = (X*Y + Y*X)/2
1335 sage: actual == expected
1337 sage: J(expected) == x*y
1340 We can change the generator prefix::
1342 sage: RealSymmetricEJA(3, prefix='q').gens()
1343 (q0, q1, q2, q3, q4, q5)
1345 Our natural basis is normalized with respect to the natural inner
1346 product unless we specify otherwise::
1348 sage: set_random_seed()
1349 sage: J = RealSymmetricEJA.random_instance()
1350 sage: all( b.norm() == 1 for b in J.gens() )
1353 Since our natural basis is normalized with respect to the natural
1354 inner product, and since we know that this algebra is an EJA, any
1355 left-multiplication operator's matrix will be symmetric because
1356 natural->EJA basis representation is an isometry and within the EJA
1357 the operator is self-adjoint by the Jordan axiom::
1359 sage: set_random_seed()
1360 sage: x = RealSymmetricEJA.random_instance().random_element()
1361 sage: x.operator().matrix().is_symmetric()
1364 We can construct the (trivial) algebra of rank zero::
1366 sage: RealSymmetricEJA(0)
1367 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
1371 def _denormalized_basis(cls
, n
, field
):
1373 Return a basis for the space of real symmetric n-by-n matrices.
1377 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1381 sage: set_random_seed()
1382 sage: n = ZZ.random_element(1,5)
1383 sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
1384 sage: all( M.is_symmetric() for M in B)
1388 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1392 for j
in range(i
+1):
1393 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1397 Sij
= Eij
+ Eij
.transpose()
1403 def _max_test_case_size():
1404 return 4 # Dimension 10
1407 def __init__(self
, n
, field
=AA
, **kwargs
):
1408 basis
= self
._denormalized
_basis
(n
, field
)
1409 super(RealSymmetricEJA
, self
).__init
__(field
, basis
, **kwargs
)
1410 self
.rank
.set_cache(n
)
1413 class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1417 Embed the n-by-n complex matrix ``M`` into the space of real
1418 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1419 bi` to the block matrix ``[[a,b],[-b,a]]``.
1423 sage: from mjo.eja.eja_algebra import \
1424 ....: ComplexMatrixEuclideanJordanAlgebra
1428 sage: F = QuadraticField(-1, 'I')
1429 sage: x1 = F(4 - 2*i)
1430 sage: x2 = F(1 + 2*i)
1433 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1434 sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1443 Embedding is a homomorphism (isomorphism, in fact)::
1445 sage: set_random_seed()
1446 sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
1447 sage: n = ZZ.random_element(n_max)
1448 sage: F = QuadraticField(-1, 'I')
1449 sage: X = random_matrix(F, n)
1450 sage: Y = random_matrix(F, n)
1451 sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
1452 sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
1453 sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1460 raise ValueError("the matrix 'M' must be square")
1462 # We don't need any adjoined elements...
1463 field
= M
.base_ring().base_ring()
1467 a
= z
.list()[0] # real part, I guess
1468 b
= z
.list()[1] # imag part, I guess
1469 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1471 return matrix
.block(field
, n
, blocks
)
1475 def real_unembed(M
):
1477 The inverse of _embed_complex_matrix().
1481 sage: from mjo.eja.eja_algebra import \
1482 ....: ComplexMatrixEuclideanJordanAlgebra
1486 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1487 ....: [-2, 1, -4, 3],
1488 ....: [ 9, 10, 11, 12],
1489 ....: [-10, 9, -12, 11] ])
1490 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
1492 [ 10*I + 9 12*I + 11]
1496 Unembedding is the inverse of embedding::
1498 sage: set_random_seed()
1499 sage: F = QuadraticField(-1, 'I')
1500 sage: M = random_matrix(F, 3)
1501 sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1502 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1508 raise ValueError("the matrix 'M' must be square")
1509 if not n
.mod(2).is_zero():
1510 raise ValueError("the matrix 'M' must be a complex embedding")
1512 # If "M" was normalized, its base ring might have roots
1513 # adjoined and they can stick around after unembedding.
1514 field
= M
.base_ring()
1515 R
= PolynomialRing(field
, 'z')
1518 # Sage doesn't know how to embed AA into QQbar, i.e. how
1519 # to adjoin sqrt(-1) to AA.
1522 F
= field
.extension(z
**2 + 1, 'I', embedding
=CLF(-1).sqrt())
1525 # Go top-left to bottom-right (reading order), converting every
1526 # 2-by-2 block we see to a single complex element.
1528 for k
in range(n
/2):
1529 for j
in range(n
/2):
1530 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1531 if submat
[0,0] != submat
[1,1]:
1532 raise ValueError('bad on-diagonal submatrix')
1533 if submat
[0,1] != -submat
[1,0]:
1534 raise ValueError('bad off-diagonal submatrix')
1535 z
= submat
[0,0] + submat
[0,1]*i
1538 return matrix(F
, n
/2, elements
)
1542 def natural_inner_product(cls
,X
,Y
):
1544 Compute a natural inner product in this algebra directly from
1549 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1553 This gives the same answer as the slow, default method implemented
1554 in :class:`MatrixEuclideanJordanAlgebra`::
1556 sage: set_random_seed()
1557 sage: J = ComplexHermitianEJA.random_instance()
1558 sage: x,y = J.random_elements(2)
1559 sage: Xe = x.natural_representation()
1560 sage: Ye = y.natural_representation()
1561 sage: X = ComplexHermitianEJA.real_unembed(Xe)
1562 sage: Y = ComplexHermitianEJA.real_unembed(Ye)
1563 sage: expected = (X*Y).trace().real()
1564 sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
1565 sage: actual == expected
1569 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/2
1572 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra
):
1574 The rank-n simple EJA consisting of complex Hermitian n-by-n
1575 matrices over the real numbers, the usual symmetric Jordan product,
1576 and the real-part-of-trace inner product. It has dimension `n^2` over
1581 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1585 In theory, our "field" can be any subfield of the reals::
1587 sage: ComplexHermitianEJA(2, RDF)
1588 Euclidean Jordan algebra of dimension 4 over Real Double Field
1589 sage: ComplexHermitianEJA(2, RR)
1590 Euclidean Jordan algebra of dimension 4 over Real Field with
1591 53 bits of precision
1595 The dimension of this algebra is `n^2`::
1597 sage: set_random_seed()
1598 sage: n_max = ComplexHermitianEJA._max_test_case_size()
1599 sage: n = ZZ.random_element(1, n_max)
1600 sage: J = ComplexHermitianEJA(n)
1601 sage: J.dimension() == n^2
1604 The Jordan multiplication is what we think it is::
1606 sage: set_random_seed()
1607 sage: J = ComplexHermitianEJA.random_instance()
1608 sage: x,y = J.random_elements(2)
1609 sage: actual = (x*y).natural_representation()
1610 sage: X = x.natural_representation()
1611 sage: Y = y.natural_representation()
1612 sage: expected = (X*Y + Y*X)/2
1613 sage: actual == expected
1615 sage: J(expected) == x*y
1618 We can change the generator prefix::
1620 sage: ComplexHermitianEJA(2, prefix='z').gens()
1623 Our natural basis is normalized with respect to the natural inner
1624 product unless we specify otherwise::
1626 sage: set_random_seed()
1627 sage: J = ComplexHermitianEJA.random_instance()
1628 sage: all( b.norm() == 1 for b in J.gens() )
1631 Since our natural basis is normalized with respect to the natural
1632 inner product, and since we know that this algebra is an EJA, any
1633 left-multiplication operator's matrix will be symmetric because
1634 natural->EJA basis representation is an isometry and within the EJA
1635 the operator is self-adjoint by the Jordan axiom::
1637 sage: set_random_seed()
1638 sage: x = ComplexHermitianEJA.random_instance().random_element()
1639 sage: x.operator().matrix().is_symmetric()
1642 We can construct the (trivial) algebra of rank zero::
1644 sage: ComplexHermitianEJA(0)
1645 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
1650 def _denormalized_basis(cls
, n
, field
):
1652 Returns a basis for the space of complex Hermitian n-by-n matrices.
1654 Why do we embed these? Basically, because all of numerical linear
1655 algebra assumes that you're working with vectors consisting of `n`
1656 entries from a field and scalars from the same field. There's no way
1657 to tell SageMath that (for example) the vectors contain complex
1658 numbers, while the scalar field is real.
1662 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1666 sage: set_random_seed()
1667 sage: n = ZZ.random_element(1,5)
1668 sage: field = QuadraticField(2, 'sqrt2')
1669 sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
1670 sage: all( M.is_symmetric() for M in B)
1674 R
= PolynomialRing(field
, 'z')
1676 F
= field
.extension(z
**2 + 1, 'I')
1679 # This is like the symmetric case, but we need to be careful:
1681 # * We want conjugate-symmetry, not just symmetry.
1682 # * The diagonal will (as a result) be real.
1686 for j
in range(i
+1):
1687 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1689 Sij
= cls
.real_embed(Eij
)
1692 # The second one has a minus because it's conjugated.
1693 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1695 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1698 # Since we embedded these, we can drop back to the "field" that we
1699 # started with instead of the complex extension "F".
1700 return ( s
.change_ring(field
) for s
in S
)
1703 def __init__(self
, n
, field
=AA
, **kwargs
):
1704 basis
= self
._denormalized
_basis
(n
,field
)
1705 super(ComplexHermitianEJA
,self
).__init
__(field
, basis
, **kwargs
)
1706 self
.rank
.set_cache(n
)
1709 class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1713 Embed the n-by-n quaternion matrix ``M`` into the space of real
1714 matrices of size 4n-by-4n by first sending each quaternion entry `z
1715 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1716 c+di],[-c + di, a-bi]]`, and then embedding those into a real
1721 sage: from mjo.eja.eja_algebra import \
1722 ....: QuaternionMatrixEuclideanJordanAlgebra
1726 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1727 sage: i,j,k = Q.gens()
1728 sage: x = 1 + 2*i + 3*j + 4*k
1729 sage: M = matrix(Q, 1, [[x]])
1730 sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1736 Embedding is a homomorphism (isomorphism, in fact)::
1738 sage: set_random_seed()
1739 sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
1740 sage: n = ZZ.random_element(n_max)
1741 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1742 sage: X = random_matrix(Q, n)
1743 sage: Y = random_matrix(Q, n)
1744 sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
1745 sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
1746 sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1751 quaternions
= M
.base_ring()
1754 raise ValueError("the matrix 'M' must be square")
1756 F
= QuadraticField(-1, 'I')
1761 t
= z
.coefficient_tuple()
1766 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1767 [-c
+ d
*i
, a
- b
*i
]])
1768 realM
= ComplexMatrixEuclideanJordanAlgebra
.real_embed(cplxM
)
1769 blocks
.append(realM
)
1771 # We should have real entries by now, so use the realest field
1772 # we've got for the return value.
1773 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1778 def real_unembed(M
):
1780 The inverse of _embed_quaternion_matrix().
1784 sage: from mjo.eja.eja_algebra import \
1785 ....: QuaternionMatrixEuclideanJordanAlgebra
1789 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1790 ....: [-2, 1, -4, 3],
1791 ....: [-3, 4, 1, -2],
1792 ....: [-4, -3, 2, 1]])
1793 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
1794 [1 + 2*i + 3*j + 4*k]
1798 Unembedding is the inverse of embedding::
1800 sage: set_random_seed()
1801 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1802 sage: M = random_matrix(Q, 3)
1803 sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1804 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1810 raise ValueError("the matrix 'M' must be square")
1811 if not n
.mod(4).is_zero():
1812 raise ValueError("the matrix 'M' must be a quaternion embedding")
1814 # Use the base ring of the matrix to ensure that its entries can be
1815 # multiplied by elements of the quaternion algebra.
1816 field
= M
.base_ring()
1817 Q
= QuaternionAlgebra(field
,-1,-1)
1820 # Go top-left to bottom-right (reading order), converting every
1821 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1824 for l
in range(n
/4):
1825 for m
in range(n
/4):
1826 submat
= ComplexMatrixEuclideanJordanAlgebra
.real_unembed(
1827 M
[4*l
:4*l
+4,4*m
:4*m
+4] )
1828 if submat
[0,0] != submat
[1,1].conjugate():
1829 raise ValueError('bad on-diagonal submatrix')
1830 if submat
[0,1] != -submat
[1,0].conjugate():
1831 raise ValueError('bad off-diagonal submatrix')
1832 z
= submat
[0,0].real()
1833 z
+= submat
[0,0].imag()*i
1834 z
+= submat
[0,1].real()*j
1835 z
+= submat
[0,1].imag()*k
1838 return matrix(Q
, n
/4, elements
)
1842 def natural_inner_product(cls
,X
,Y
):
1844 Compute a natural inner product in this algebra directly from
1849 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1853 This gives the same answer as the slow, default method implemented
1854 in :class:`MatrixEuclideanJordanAlgebra`::
1856 sage: set_random_seed()
1857 sage: J = QuaternionHermitianEJA.random_instance()
1858 sage: x,y = J.random_elements(2)
1859 sage: Xe = x.natural_representation()
1860 sage: Ye = y.natural_representation()
1861 sage: X = QuaternionHermitianEJA.real_unembed(Xe)
1862 sage: Y = QuaternionHermitianEJA.real_unembed(Ye)
1863 sage: expected = (X*Y).trace().coefficient_tuple()[0]
1864 sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye)
1865 sage: actual == expected
1869 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/4
1872 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra
):
1874 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1875 matrices, the usual symmetric Jordan product, and the
1876 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1881 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1885 In theory, our "field" can be any subfield of the reals::
1887 sage: QuaternionHermitianEJA(2, RDF)
1888 Euclidean Jordan algebra of dimension 6 over Real Double Field
1889 sage: QuaternionHermitianEJA(2, RR)
1890 Euclidean Jordan algebra of dimension 6 over Real Field with
1891 53 bits of precision
1895 The dimension of this algebra is `2*n^2 - n`::
1897 sage: set_random_seed()
1898 sage: n_max = QuaternionHermitianEJA._max_test_case_size()
1899 sage: n = ZZ.random_element(1, n_max)
1900 sage: J = QuaternionHermitianEJA(n)
1901 sage: J.dimension() == 2*(n^2) - n
1904 The Jordan multiplication is what we think it is::
1906 sage: set_random_seed()
1907 sage: J = QuaternionHermitianEJA.random_instance()
1908 sage: x,y = J.random_elements(2)
1909 sage: actual = (x*y).natural_representation()
1910 sage: X = x.natural_representation()
1911 sage: Y = y.natural_representation()
1912 sage: expected = (X*Y + Y*X)/2
1913 sage: actual == expected
1915 sage: J(expected) == x*y
1918 We can change the generator prefix::
1920 sage: QuaternionHermitianEJA(2, prefix='a').gens()
1921 (a0, a1, a2, a3, a4, a5)
1923 Our natural basis is normalized with respect to the natural inner
1924 product unless we specify otherwise::
1926 sage: set_random_seed()
1927 sage: J = QuaternionHermitianEJA.random_instance()
1928 sage: all( b.norm() == 1 for b in J.gens() )
1931 Since our natural basis is normalized with respect to the natural
1932 inner product, and since we know that this algebra is an EJA, any
1933 left-multiplication operator's matrix will be symmetric because
1934 natural->EJA basis representation is an isometry and within the EJA
1935 the operator is self-adjoint by the Jordan axiom::
1937 sage: set_random_seed()
1938 sage: x = QuaternionHermitianEJA.random_instance().random_element()
1939 sage: x.operator().matrix().is_symmetric()
1942 We can construct the (trivial) algebra of rank zero::
1944 sage: QuaternionHermitianEJA(0)
1945 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
1949 def _denormalized_basis(cls
, n
, field
):
1951 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1953 Why do we embed these? Basically, because all of numerical
1954 linear algebra assumes that you're working with vectors consisting
1955 of `n` entries from a field and scalars from the same field. There's
1956 no way to tell SageMath that (for example) the vectors contain
1957 complex numbers, while the scalar field is real.
1961 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1965 sage: set_random_seed()
1966 sage: n = ZZ.random_element(1,5)
1967 sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
1968 sage: all( M.is_symmetric() for M in B )
1972 Q
= QuaternionAlgebra(QQ
,-1,-1)
1975 # This is like the symmetric case, but we need to be careful:
1977 # * We want conjugate-symmetry, not just symmetry.
1978 # * The diagonal will (as a result) be real.
1982 for j
in range(i
+1):
1983 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1985 Sij
= cls
.real_embed(Eij
)
1988 # The second, third, and fourth ones have a minus
1989 # because they're conjugated.
1990 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1992 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1994 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
1996 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
1999 # Since we embedded these, we can drop back to the "field" that we
2000 # started with instead of the quaternion algebra "Q".
2001 return ( s
.change_ring(field
) for s
in S
)
2004 def __init__(self
, n
, field
=AA
, **kwargs
):
2005 basis
= self
._denormalized
_basis
(n
,field
)
2006 super(QuaternionHermitianEJA
,self
).__init
__(field
, basis
, **kwargs
)
2007 self
.rank
.set_cache(n
)
2010 class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2012 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2013 with the half-trace inner product and jordan product ``x*y =
2014 (x0*y0 + <B*x_bar,y_bar>, x0*y_bar + y0*x_bar)`` where ``B`` is a
2015 symmetric positive-definite "bilinear form" matrix. It has
2016 dimension `n` over the reals, and reduces to the ``JordanSpinEJA``
2017 when ``B`` is the identity matrix of order ``n-1``.
2021 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
2022 ....: JordanSpinEJA)
2026 When no bilinear form is specified, the identity matrix is used,
2027 and the resulting algebra is the Jordan spin algebra::
2029 sage: J0 = BilinearFormEJA(3)
2030 sage: J1 = JordanSpinEJA(3)
2031 sage: J0.multiplication_table() == J0.multiplication_table()
2036 We can create a zero-dimensional algebra::
2038 sage: J = BilinearFormEJA(0)
2042 We can check the multiplication condition given in the Jordan, von
2043 Neumann, and Wigner paper (and also discussed on my "On the
2044 symmetry..." paper). Note that this relies heavily on the standard
2045 choice of basis, as does anything utilizing the bilinear form matrix::
2047 sage: set_random_seed()
2048 sage: n = ZZ.random_element(5)
2049 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
2050 sage: B = M.transpose()*M
2051 sage: J = BilinearFormEJA(n, B=B)
2052 sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
2053 sage: V = J.vector_space()
2054 sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
2055 ....: for ei in eis ]
2056 sage: actual = [ sis[i]*sis[j]
2057 ....: for i in range(n-1)
2058 ....: for j in range(n-1) ]
2059 sage: expected = [ J.one() if i == j else J.zero()
2060 ....: for i in range(n-1)
2061 ....: for j in range(n-1) ]
2062 sage: actual == expected
2065 def __init__(self
, n
, field
=AA
, B
=None, **kwargs
):
2067 self
._B
= matrix
.identity(field
, max(0,n
-1))
2071 V
= VectorSpace(field
, n
)
2072 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
2081 z0
= x0
*y0
+ (self
._B
*xbar
).inner_product(ybar
)
2082 zbar
= y0
*xbar
+ x0
*ybar
2083 z
= V([z0
] + zbar
.list())
2084 mult_table
[i
][j
] = z
2086 # The rank of this algebra is two, unless we're in a
2087 # one-dimensional ambient space (because the rank is bounded
2088 # by the ambient dimension).
2089 fdeja
= super(BilinearFormEJA
, self
)
2090 fdeja
.__init
__(field
, mult_table
, **kwargs
)
2091 self
.rank
.set_cache(min(n
,2))
2093 def inner_product(self
, x
, y
):
2095 Half of the trace inner product.
2097 This is defined so that the special case of the Jordan spin
2098 algebra gets the usual inner product.
2102 sage: from mjo.eja.eja_algebra import BilinearFormEJA
2106 Ensure that this is one-half of the trace inner-product when
2107 the algebra isn't just the reals (when ``n`` isn't one). This
2108 is in Faraut and Koranyi, and also my "On the symmetry..."
2111 sage: set_random_seed()
2112 sage: n = ZZ.random_element(2,5)
2113 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
2114 sage: B = M.transpose()*M
2115 sage: J = BilinearFormEJA(n, B=B)
2116 sage: x = J.random_element()
2117 sage: y = J.random_element()
2118 sage: x.inner_product(y) == (x*y).trace()/2
2122 xvec
= x
.to_vector()
2124 yvec
= y
.to_vector()
2126 return x
[0]*y
[0] + (self
._B
*xbar
).inner_product(ybar
)
2129 class JordanSpinEJA(BilinearFormEJA
):
2131 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2132 with the usual inner product and jordan product ``x*y =
2133 (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2138 sage: from mjo.eja.eja_algebra import JordanSpinEJA
2142 This multiplication table can be verified by hand::
2144 sage: J = JordanSpinEJA(4)
2145 sage: e0,e1,e2,e3 = J.gens()
2161 We can change the generator prefix::
2163 sage: JordanSpinEJA(2, prefix='B').gens()
2168 Ensure that we have the usual inner product on `R^n`::
2170 sage: set_random_seed()
2171 sage: J = JordanSpinEJA.random_instance()
2172 sage: x,y = J.random_elements(2)
2173 sage: X = x.natural_representation()
2174 sage: Y = y.natural_representation()
2175 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
2179 def __init__(self
, n
, field
=AA
, **kwargs
):
2180 # This is a special case of the BilinearFormEJA with the identity
2181 # matrix as its bilinear form.
2182 return super(JordanSpinEJA
, self
).__init
__(n
, field
, **kwargs
)
2185 class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2187 The trivial Euclidean Jordan algebra consisting of only a zero element.
2191 sage: from mjo.eja.eja_algebra import TrivialEJA
2195 sage: J = TrivialEJA()
2202 sage: 7*J.one()*12*J.one()
2204 sage: J.one().inner_product(J.one())
2206 sage: J.one().norm()
2208 sage: J.one().subalgebra_generated_by()
2209 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
2214 def __init__(self
, field
=AA
, **kwargs
):
2216 fdeja
= super(TrivialEJA
, self
)
2217 # The rank is zero using my definition, namely the dimension of the
2218 # largest subalgebra generated by any element.
2219 fdeja
.__init
__(field
, mult_table
, **kwargs
)
2220 self
.rank
.set_cache(0)
2223 class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2225 The external (orthogonal) direct sum of two other Euclidean Jordan
2226 algebras. Essentially the Cartesian product of its two factors.
2227 Every Euclidean Jordan algebra decomposes into an orthogonal
2228 direct sum of simple Euclidean Jordan algebras, so no generality
2229 is lost by providing only this construction.
2233 sage: from mjo.eja.eja_algebra import (HadamardEJA,
2234 ....: RealSymmetricEJA,
2239 sage: J1 = HadamardEJA(2)
2240 sage: J2 = RealSymmetricEJA(3)
2241 sage: J = DirectSumEJA(J1,J2)
2248 def __init__(self
, J1
, J2
, field
=AA
, **kwargs
):
2252 V
= VectorSpace(field
, n
)
2253 mult_table
= [ [ V
.zero() for j
in range(n
) ]
2257 p
= (J1
.monomial(i
)*J1
.monomial(j
)).to_vector()
2258 mult_table
[i
][j
] = V(p
.list() + [field
.zero()]*n2
)
2262 p
= (J2
.monomial(i
)*J2
.monomial(j
)).to_vector()
2263 mult_table
[n1
+i
][n1
+j
] = V([field
.zero()]*n1
+ p
.list())
2265 fdeja
= super(DirectSumEJA
, self
)
2266 fdeja
.__init
__(field
, mult_table
, **kwargs
)
2267 self
.rank
.set_cache(J1
.rank() + J2
.rank())