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
65 sage: from mjo.eja.eja_algebra import (
66 ....: FiniteDimensionalEuclideanJordanAlgebra,
72 By definition, Jordan multiplication commutes::
74 sage: set_random_seed()
75 sage: J = random_eja()
76 sage: x,y = J.random_elements(2)
82 The ``field`` we're given must be real with ``check_field=True``::
84 sage: JordanSpinEJA(2,QQbar)
85 Traceback (most recent call last):
87 ValueError: scalar field is not real
89 The multiplication table must be square with ``check_axioms=True``::
91 sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),()))
92 Traceback (most recent call last):
94 ValueError: multiplication table is not square
98 if not field
.is_subring(RR
):
99 # Note: this does return true for the real algebraic
100 # field, the rationals, and any quadratic field where
101 # we've specified a real embedding.
102 raise ValueError("scalar field is not real")
104 # The multiplication table had better be square
107 if not all( len(l
) == n
for l
in mult_table
):
108 raise ValueError("multiplication table is not square")
110 self
._natural
_basis
= natural_basis
113 category
= MagmaticAlgebras(field
).FiniteDimensional()
114 category
= category
.WithBasis().Unital()
116 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
121 self
.print_options(bracket
='')
123 # The multiplication table we're given is necessarily in terms
124 # of vectors, because we don't have an algebra yet for
125 # anything to be an element of. However, it's faster in the
126 # long run to have the multiplication table be in terms of
127 # algebra elements. We do this after calling the superclass
128 # constructor so that from_vector() knows what to do.
129 self
._multiplication
_table
= [
130 list(map(lambda x
: self
.from_vector(x
), ls
))
135 if not self
._is
_commutative
():
136 raise ValueError("algebra is not commutative")
137 if not self
._is
_jordanian
():
138 raise ValueError("Jordan identity does not hold")
139 if not self
._inner
_product
_is
_associative
():
140 raise ValueError("inner product is not associative")
142 def _element_constructor_(self
, elt
):
144 Construct an element of this algebra from its natural
147 This gets called only after the parent element _call_ method
148 fails to find a coercion for the argument.
152 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
154 ....: RealSymmetricEJA)
158 The identity in `S^n` is converted to the identity in the EJA::
160 sage: J = RealSymmetricEJA(3)
161 sage: I = matrix.identity(QQ,3)
162 sage: J(I) == J.one()
165 This skew-symmetric matrix can't be represented in the EJA::
167 sage: J = RealSymmetricEJA(3)
168 sage: A = matrix(QQ,3, lambda i,j: i-j)
170 Traceback (most recent call last):
172 ArithmeticError: vector is not in free module
176 Ensure that we can convert any element of the two non-matrix
177 simple algebras (whose natural representations are their usual
178 vector representations) back and forth faithfully::
180 sage: set_random_seed()
181 sage: J = HadamardEJA.random_instance()
182 sage: x = J.random_element()
183 sage: J(x.to_vector().column()) == x
185 sage: J = JordanSpinEJA.random_instance()
186 sage: x = J.random_element()
187 sage: J(x.to_vector().column()) == x
191 msg
= "not a naturally-represented algebra element"
193 # The superclass implementation of random_element()
194 # needs to be able to coerce "0" into the algebra.
196 elif elt
in self
.base_ring():
197 # Ensure that no base ring -> algebra coercion is performed
198 # by this method. There's some stupidity in sage that would
199 # otherwise propagate to this method; for example, sage thinks
200 # that the integer 3 belongs to the space of 2-by-2 matrices.
201 raise ValueError(msg
)
203 natural_basis
= self
.natural_basis()
204 basis_space
= natural_basis
[0].matrix_space()
205 if elt
not in basis_space
:
206 raise ValueError(msg
)
208 # Thanks for nothing! Matrix spaces aren't vector spaces in
209 # Sage, so we have to figure out its natural-basis coordinates
210 # ourselves. We use the basis space's ring instead of the
211 # element's ring because the basis space might be an algebraic
212 # closure whereas the base ring of the 3-by-3 identity matrix
213 # could be QQ instead of QQbar.
214 V
= VectorSpace(basis_space
.base_ring(), elt
.nrows()*elt
.ncols())
215 W
= V
.span_of_basis( _mat2vec(s
) for s
in natural_basis
)
216 coords
= W
.coordinate_vector(_mat2vec(elt
))
217 return self
.from_vector(coords
)
220 def _max_test_case_size():
222 Return an integer "size" that is an upper bound on the size of
223 this algebra when it is used in a random test
224 case. Unfortunately, the term "size" is quite vague -- when
225 dealing with `R^n` under either the Hadamard or Jordan spin
226 product, the "size" refers to the dimension `n`. When dealing
227 with a matrix algebra (real symmetric or complex/quaternion
228 Hermitian), it refers to the size of the matrix, which is
229 far less than the dimension of the underlying vector space.
231 We default to five in this class, which is safe in `R^n`. The
232 matrix algebra subclasses (or any class where the "size" is
233 interpreted to be far less than the dimension) should override
234 with a smaller number.
240 Return a string representation of ``self``.
244 sage: from mjo.eja.eja_algebra import JordanSpinEJA
248 Ensure that it says what we think it says::
250 sage: JordanSpinEJA(2, field=AA)
251 Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
252 sage: JordanSpinEJA(3, field=RDF)
253 Euclidean Jordan algebra of dimension 3 over Real Double Field
256 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
257 return fmt
.format(self
.dimension(), self
.base_ring())
259 def product_on_basis(self
, i
, j
):
260 return self
._multiplication
_table
[i
][j
]
262 def _is_commutative(self
):
264 Whether or not this algebra's multiplication table is commutative.
266 This method should of course always return ``True``, unless
267 this algebra was constructed with ``check_axioms=False`` and
268 passed an invalid multiplication table.
270 return all( self
.product_on_basis(i
,j
) == self
.product_on_basis(i
,j
)
271 for i
in range(self
.dimension())
272 for j
in range(self
.dimension()) )
274 def _is_jordanian(self
):
276 Whether or not this algebra's multiplication table respects the
277 Jordan identity `(x^{2})(xy) = x(x^{2}y)`.
279 We only check one arrangement of `x` and `y`, so for a
280 ``True`` result to be truly true, you should also check
281 :meth:`_is_commutative`. This method should of course always
282 return ``True``, unless this algebra was constructed with
283 ``check_axioms=False`` and passed an invalid multiplication table.
285 return all( (self
.monomial(i
)**2)*(self
.monomial(i
)*self
.monomial(j
))
287 (self
.monomial(i
))*((self
.monomial(i
)**2)*self
.monomial(j
))
288 for i
in range(self
.dimension())
289 for j
in range(self
.dimension()) )
291 def _inner_product_is_associative(self
):
293 Return whether or not this algebra's inner product `B` is
294 associative; that is, whether or not `B(xy,z) = B(x,yz)`.
296 This method should of course always return ``True``, unless
297 this algebra was constructed with ``check_axioms=False`` and
298 passed an invalid multiplication table.
301 # Used to check whether or not something is zero in an inexact
302 # ring. This number is sufficient to allow the construction of
303 # QuaternionHermitianEJA(2, RDF) with check_axioms=True.
306 for i
in range(self
.dimension()):
307 for j
in range(self
.dimension()):
308 for k
in range(self
.dimension()):
312 diff
= (x
*y
).inner_product(z
) - x
.inner_product(y
*z
)
314 if self
.base_ring().is_exact():
318 if diff
.abs() > epsilon
:
324 def characteristic_polynomial_of(self
):
326 Return the algebra's "characteristic polynomial of" function,
327 which is itself a multivariate polynomial that, when evaluated
328 at the coordinates of some algebra element, returns that
329 element's characteristic polynomial.
331 The resulting polynomial has `n+1` variables, where `n` is the
332 dimension of this algebra. The first `n` variables correspond to
333 the coordinates of an algebra element: when evaluated at the
334 coordinates of an algebra element with respect to a certain
335 basis, the result is a univariate polynomial (in the one
336 remaining variable ``t``), namely the characteristic polynomial
341 sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA
345 The characteristic polynomial in the spin algebra is given in
346 Alizadeh, Example 11.11::
348 sage: J = JordanSpinEJA(3)
349 sage: p = J.characteristic_polynomial_of(); p
350 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
351 sage: xvec = J.one().to_vector()
355 By definition, the characteristic polynomial is a monic
356 degree-zero polynomial in a rank-zero algebra. Note that
357 Cayley-Hamilton is indeed satisfied since the polynomial
358 ``1`` evaluates to the identity element of the algebra on
361 sage: J = TrivialEJA()
362 sage: J.characteristic_polynomial_of()
369 # The list of coefficient polynomials a_0, a_1, a_2, ..., a_(r-1).
370 a
= self
._charpoly
_coefficients
()
372 # We go to a bit of trouble here to reorder the
373 # indeterminates, so that it's easier to evaluate the
374 # characteristic polynomial at x's coordinates and get back
375 # something in terms of t, which is what we want.
376 S
= PolynomialRing(self
.base_ring(),'t')
380 S
= PolynomialRing(S
, R
.variable_names())
383 return (t
**r
+ sum( a
[k
]*(t
**k
) for k
in range(r
) ))
386 def inner_product(self
, x
, y
):
388 The inner product associated with this Euclidean Jordan algebra.
390 Defaults to the trace inner product, but can be overridden by
391 subclasses if they are sure that the necessary properties are
396 sage: from mjo.eja.eja_algebra import random_eja
400 Our inner product is "associative," which means the following for
401 a symmetric bilinear form::
403 sage: set_random_seed()
404 sage: J = random_eja()
405 sage: x,y,z = J.random_elements(3)
406 sage: (x*y).inner_product(z) == y.inner_product(x*z)
410 X
= x
.natural_representation()
411 Y
= y
.natural_representation()
412 return self
.natural_inner_product(X
,Y
)
415 def is_trivial(self
):
417 Return whether or not this algebra is trivial.
419 A trivial algebra contains only the zero element.
423 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
428 sage: J = ComplexHermitianEJA(3)
434 sage: J = TrivialEJA()
439 return self
.dimension() == 0
442 def multiplication_table(self
):
444 Return a visual representation of this algebra's multiplication
445 table (on basis elements).
449 sage: from mjo.eja.eja_algebra import JordanSpinEJA
453 sage: J = JordanSpinEJA(4)
454 sage: J.multiplication_table()
455 +----++----+----+----+----+
456 | * || e0 | e1 | e2 | e3 |
457 +====++====+====+====+====+
458 | e0 || e0 | e1 | e2 | e3 |
459 +----++----+----+----+----+
460 | e1 || e1 | e0 | 0 | 0 |
461 +----++----+----+----+----+
462 | e2 || e2 | 0 | e0 | 0 |
463 +----++----+----+----+----+
464 | e3 || e3 | 0 | 0 | e0 |
465 +----++----+----+----+----+
468 M
= list(self
._multiplication
_table
) # copy
469 for i
in range(len(M
)):
470 # M had better be "square"
471 M
[i
] = [self
.monomial(i
)] + M
[i
]
472 M
= [["*"] + list(self
.gens())] + M
473 return table(M
, header_row
=True, header_column
=True, frame
=True)
476 def natural_basis(self
):
478 Return a more-natural representation of this algebra's basis.
480 Every finite-dimensional Euclidean Jordan Algebra is a direct
481 sum of five simple algebras, four of which comprise Hermitian
482 matrices. This method returns the original "natural" basis
483 for our underlying vector space. (Typically, the natural basis
484 is used to construct the multiplication table in the first place.)
486 Note that this will always return a matrix. The standard basis
487 in `R^n` will be returned as `n`-by-`1` column matrices.
491 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
492 ....: RealSymmetricEJA)
496 sage: J = RealSymmetricEJA(2)
498 Finite family {0: e0, 1: e1, 2: e2}
499 sage: J.natural_basis()
501 [1 0] [ 0 0.7071067811865475?] [0 0]
502 [0 0], [0.7071067811865475? 0], [0 1]
507 sage: J = JordanSpinEJA(2)
509 Finite family {0: e0, 1: e1}
510 sage: J.natural_basis()
517 if self
._natural
_basis
is None:
518 M
= self
.natural_basis_space()
519 return tuple( M(b
.to_vector()) for b
in self
.basis() )
521 return self
._natural
_basis
524 def natural_basis_space(self
):
526 Return the matrix space in which this algebra's natural basis
529 Generally this will be an `n`-by-`1` column-vector space,
530 except when the algebra is trivial. There it's `n`-by-`n`
531 (where `n` is zero), to ensure that two elements of the
532 natural basis space (empty matrices) can be multiplied.
534 if self
.is_trivial():
535 return MatrixSpace(self
.base_ring(), 0)
536 elif self
._natural
_basis
is None or len(self
._natural
_basis
) == 0:
537 return MatrixSpace(self
.base_ring(), self
.dimension(), 1)
539 return self
._natural
_basis
[0].matrix_space()
543 def natural_inner_product(X
,Y
):
545 Compute the inner product of two naturally-represented elements.
547 For example in the real symmetric matrix EJA, this will compute
548 the trace inner-product of two n-by-n symmetric matrices. The
549 default should work for the real cartesian product EJA, the
550 Jordan spin EJA, and the real symmetric matrices. The others
551 will have to be overridden.
553 return (X
.conjugate_transpose()*Y
).trace()
559 Return the unit element of this algebra.
563 sage: from mjo.eja.eja_algebra import (HadamardEJA,
568 sage: J = HadamardEJA(5)
570 e0 + e1 + e2 + e3 + e4
574 The identity element acts like the identity::
576 sage: set_random_seed()
577 sage: J = random_eja()
578 sage: x = J.random_element()
579 sage: J.one()*x == x and x*J.one() == x
582 The matrix of the unit element's operator is the identity::
584 sage: set_random_seed()
585 sage: J = random_eja()
586 sage: actual = J.one().operator().matrix()
587 sage: expected = matrix.identity(J.base_ring(), J.dimension())
588 sage: actual == expected
592 # We can brute-force compute the matrices of the operators
593 # that correspond to the basis elements of this algebra.
594 # If some linear combination of those basis elements is the
595 # algebra identity, then the same linear combination of
596 # their matrices has to be the identity matrix.
598 # Of course, matrices aren't vectors in sage, so we have to
599 # appeal to the "long vectors" isometry.
600 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
602 # Now we use basis linear algebra to find the coefficients,
603 # of the matrices-as-vectors-linear-combination, which should
604 # work for the original algebra basis too.
605 A
= matrix
.column(self
.base_ring(), oper_vecs
)
607 # We used the isometry on the left-hand side already, but we
608 # still need to do it for the right-hand side. Recall that we
609 # wanted something that summed to the identity matrix.
610 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
612 # Now if there's an identity element in the algebra, this should work.
613 coeffs
= A
.solve_right(b
)
614 return self
.linear_combination(zip(self
.gens(), coeffs
))
617 def peirce_decomposition(self
, c
):
619 The Peirce decomposition of this algebra relative to the
622 In the future, this can be extended to a complete system of
623 orthogonal idempotents.
627 - ``c`` -- an idempotent of this algebra.
631 A triple (J0, J5, J1) containing two subalgebras and one subspace
634 - ``J0`` -- the algebra on the eigenspace of ``c.operator()``
635 corresponding to the eigenvalue zero.
637 - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()``
638 corresponding to the eigenvalue one-half.
640 - ``J1`` -- the algebra on the eigenspace of ``c.operator()``
641 corresponding to the eigenvalue one.
643 These are the only possible eigenspaces for that operator, and this
644 algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are
645 orthogonal, and are subalgebras of this algebra with the appropriate
650 sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA
654 The canonical example comes from the symmetric matrices, which
655 decompose into diagonal and off-diagonal parts::
657 sage: J = RealSymmetricEJA(3)
658 sage: C = matrix(QQ, [ [1,0,0],
662 sage: J0,J5,J1 = J.peirce_decomposition(c)
664 Euclidean Jordan algebra of dimension 1...
666 Vector space of degree 6 and dimension 2...
668 Euclidean Jordan algebra of dimension 3...
669 sage: J0.one().natural_representation()
673 sage: orig_df = AA.options.display_format
674 sage: AA.options.display_format = 'radical'
675 sage: J.from_vector(J5.basis()[0]).natural_representation()
679 sage: J.from_vector(J5.basis()[1]).natural_representation()
683 sage: AA.options.display_format = orig_df
684 sage: J1.one().natural_representation()
691 Every algebra decomposes trivially with respect to its identity
694 sage: set_random_seed()
695 sage: J = random_eja()
696 sage: J0,J5,J1 = J.peirce_decomposition(J.one())
697 sage: J0.dimension() == 0 and J5.dimension() == 0
699 sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
702 The decomposition is into eigenspaces, and its components are
703 therefore necessarily orthogonal. Moreover, the identity
704 elements in the two subalgebras are the projections onto their
705 respective subspaces of the superalgebra's identity element::
707 sage: set_random_seed()
708 sage: J = random_eja()
709 sage: x = J.random_element()
710 sage: if not J.is_trivial():
711 ....: while x.is_nilpotent():
712 ....: x = J.random_element()
713 sage: c = x.subalgebra_idempotent()
714 sage: J0,J5,J1 = J.peirce_decomposition(c)
716 sage: for (w,y,z) in zip(J0.basis(), J5.basis(), J1.basis()):
717 ....: w = w.superalgebra_element()
718 ....: y = J.from_vector(y)
719 ....: z = z.superalgebra_element()
720 ....: ipsum += w.inner_product(y).abs()
721 ....: ipsum += w.inner_product(z).abs()
722 ....: ipsum += y.inner_product(z).abs()
725 sage: J1(c) == J1.one()
727 sage: J0(J.one() - c) == J0.one()
731 if not c
.is_idempotent():
732 raise ValueError("element is not idempotent: %s" % c
)
734 # Default these to what they should be if they turn out to be
735 # trivial, because eigenspaces_left() won't return eigenvalues
736 # corresponding to trivial spaces (e.g. it returns only the
737 # eigenspace corresponding to lambda=1 if you take the
738 # decomposition relative to the identity element).
739 trivial
= FiniteDimensionalEuclideanJordanSubalgebra(self
, ())
740 J0
= trivial
# eigenvalue zero
741 J5
= VectorSpace(self
.base_ring(), 0) # eigenvalue one-half
742 J1
= trivial
# eigenvalue one
744 for (eigval
, eigspace
) in c
.operator().matrix().right_eigenspaces():
745 if eigval
== ~
(self
.base_ring()(2)):
748 gens
= tuple( self
.from_vector(b
) for b
in eigspace
.basis() )
749 subalg
= FiniteDimensionalEuclideanJordanSubalgebra(self
,
757 raise ValueError("unexpected eigenvalue: %s" % eigval
)
762 def random_element(self
, thorough
=False):
764 Return a random element of this algebra.
766 Our algebra superclass method only returns a linear
767 combination of at most two basis elements. We instead
768 want the vector space "random element" method that
769 returns a more diverse selection.
773 - ``thorough`` -- (boolean; default False) whether or not we
774 should generate irrational coefficients for the random
775 element when our base ring is irrational; this slows the
776 algebra operations to a crawl, but any truly random method
780 # For a general base ring... maybe we can trust this to do the
781 # right thing? Unlikely, but.
782 V
= self
.vector_space()
783 v
= V
.random_element()
785 if self
.base_ring() is AA
:
786 # The "random element" method of the algebraic reals is
787 # stupid at the moment, and only returns integers between
788 # -2 and 2, inclusive:
790 # https://trac.sagemath.org/ticket/30875
792 # Instead, we implement our own "random vector" method,
793 # and then coerce that into the algebra. We use the vector
794 # space degree here instead of the dimension because a
795 # subalgebra could (for example) be spanned by only two
796 # vectors, each with five coordinates. We need to
797 # generate all five coordinates.
799 v
*= QQbar
.random_element().real()
801 v
*= QQ
.random_element()
803 return self
.from_vector(V
.coordinate_vector(v
))
805 def random_elements(self
, count
, thorough
=False):
807 Return ``count`` random elements as a tuple.
811 - ``thorough`` -- (boolean; default False) whether or not we
812 should generate irrational coefficients for the random
813 elements when our base ring is irrational; this slows the
814 algebra operations to a crawl, but any truly random method
819 sage: from mjo.eja.eja_algebra import JordanSpinEJA
823 sage: J = JordanSpinEJA(3)
824 sage: x,y,z = J.random_elements(3)
825 sage: all( [ x in J, y in J, z in J ])
827 sage: len( J.random_elements(10) ) == 10
831 return tuple( self
.random_element(thorough
)
832 for idx
in range(count
) )
835 def random_instance(cls
, field
=AA
, **kwargs
):
837 Return a random instance of this type of algebra.
839 Beware, this will crash for "most instances" because the
840 constructor below looks wrong.
842 if cls
is TrivialEJA
:
843 # The TrivialEJA class doesn't take an "n" argument because
847 n
= ZZ
.random_element(cls
._max
_test
_case
_size
() + 1)
848 return cls(n
, field
, **kwargs
)
851 def _charpoly_coefficients(self
):
853 The `r` polynomial coefficients of the "characteristic polynomial
857 var_names
= [ "X" + str(z
) for z
in range(1,n
+1) ]
858 R
= PolynomialRing(self
.base_ring(), var_names
)
860 F
= R
.fraction_field()
863 # From a result in my book, these are the entries of the
864 # basis representation of L_x.
865 return sum( vars[k
]*self
.monomial(k
).operator().matrix()[i
,j
]
868 L_x
= matrix(F
, n
, n
, L_x_i_j
)
871 if self
.rank
.is_in_cache():
873 # There's no need to pad the system with redundant
874 # columns if we *know* they'll be redundant.
877 # Compute an extra power in case the rank is equal to
878 # the dimension (otherwise, we would stop at x^(r-1)).
879 x_powers
= [ (L_x
**k
)*self
.one().to_vector()
880 for k
in range(n
+1) ]
881 A
= matrix
.column(F
, x_powers
[:n
])
882 AE
= A
.extended_echelon_form()
889 # The theory says that only the first "r" coefficients are
890 # nonzero, and they actually live in the original polynomial
891 # ring and not the fraction field. We negate them because
892 # in the actual characteristic polynomial, they get moved
893 # to the other side where x^r lives.
894 return -A_rref
.solve_right(E
*b
).change_ring(R
)[:r
]
899 Return the rank of this EJA.
901 This is a cached method because we know the rank a priori for
902 all of the algebras we can construct. Thus we can avoid the
903 expensive ``_charpoly_coefficients()`` call unless we truly
904 need to compute the whole characteristic polynomial.
908 sage: from mjo.eja.eja_algebra import (HadamardEJA,
910 ....: RealSymmetricEJA,
911 ....: ComplexHermitianEJA,
912 ....: QuaternionHermitianEJA,
917 The rank of the Jordan spin algebra is always two::
919 sage: JordanSpinEJA(2).rank()
921 sage: JordanSpinEJA(3).rank()
923 sage: JordanSpinEJA(4).rank()
926 The rank of the `n`-by-`n` Hermitian real, complex, or
927 quaternion matrices is `n`::
929 sage: RealSymmetricEJA(4).rank()
931 sage: ComplexHermitianEJA(3).rank()
933 sage: QuaternionHermitianEJA(2).rank()
938 Ensure that every EJA that we know how to construct has a
939 positive integer rank, unless the algebra is trivial in
940 which case its rank will be zero::
942 sage: set_random_seed()
943 sage: J = random_eja()
947 sage: r > 0 or (r == 0 and J.is_trivial())
950 Ensure that computing the rank actually works, since the ranks
951 of all simple algebras are known and will be cached by default::
953 sage: J = HadamardEJA(4)
954 sage: J.rank.clear_cache()
960 sage: J = JordanSpinEJA(4)
961 sage: J.rank.clear_cache()
967 sage: J = RealSymmetricEJA(3)
968 sage: J.rank.clear_cache()
974 sage: J = ComplexHermitianEJA(2)
975 sage: J.rank.clear_cache()
981 sage: J = QuaternionHermitianEJA(2)
982 sage: J.rank.clear_cache()
986 return len(self
._charpoly
_coefficients
())
989 def vector_space(self
):
991 Return the vector space that underlies this algebra.
995 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
999 sage: J = RealSymmetricEJA(2)
1000 sage: J.vector_space()
1001 Vector space of dimension 3 over...
1004 return self
.zero().to_vector().parent().ambient_vector_space()
1007 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
1011 def random_eja(field
=AA
):
1013 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1017 sage: from mjo.eja.eja_algebra import random_eja
1022 Euclidean Jordan algebra of dimension...
1025 classname
= choice([TrivialEJA
,
1029 ComplexHermitianEJA
,
1030 QuaternionHermitianEJA
])
1031 return classname
.random_instance(field
=field
)
1036 class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
1038 Algebras whose basis consists of vectors with rational
1039 entries. Equivalently, algebras whose multiplication tables
1040 contain only rational coefficients.
1042 When an EJA has a basis that can be made rational, we can speed up
1043 the computation of its characteristic polynomial by doing it over
1044 ``QQ``. All of the named EJA constructors that we provide fall
1048 def _charpoly_coefficients(self
):
1050 Override the parent method with something that tries to compute
1051 over a faster (non-extension) field.
1053 if self
.base_ring() is QQ
:
1054 # There's no need to construct *another* algebra over the
1055 # rationals if this one is already over the rationals.
1056 superclass
= super(RationalBasisEuclideanJordanAlgebra
, self
)
1057 return superclass
._charpoly
_coefficients
()
1060 map(lambda x
: x
.to_vector(), ls
)
1061 for ls
in self
._multiplication
_table
1064 # Do the computation over the rationals. The answer will be
1065 # the same, because our basis coordinates are (essentially)
1067 J
= FiniteDimensionalEuclideanJordanAlgebra(QQ
,
1071 return J
._charpoly
_coefficients
()
1074 class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
1076 def _max_test_case_size():
1077 # Play it safe, since this will be squared and the underlying
1078 # field can have dimension 4 (quaternions) too.
1081 def __init__(self
, field
, basis
, normalize_basis
=True, **kwargs
):
1083 Compared to the superclass constructor, we take a basis instead of
1084 a multiplication table because the latter can be computed in terms
1085 of the former when the product is known (like it is here).
1087 # Used in this class's fast _charpoly_coefficients() override.
1088 self
._basis
_normalizers
= None
1090 # We're going to loop through this a few times, so now's a good
1091 # time to ensure that it isn't a generator expression.
1092 basis
= tuple(basis
)
1094 if len(basis
) > 1 and normalize_basis
:
1095 # We'll need sqrt(2) to normalize the basis, and this
1096 # winds up in the multiplication table, so the whole
1097 # algebra needs to be over the field extension.
1098 R
= PolynomialRing(field
, 'z')
1101 if p
.is_irreducible():
1102 field
= field
.extension(p
, 'sqrt2', embedding
=RLF(2).sqrt())
1103 basis
= tuple( s
.change_ring(field
) for s
in basis
)
1104 self
._basis
_normalizers
= tuple(
1105 ~
(self
.natural_inner_product(s
,s
).sqrt()) for s
in basis
)
1106 basis
= tuple(s
*c
for (s
,c
) in zip(basis
,self
._basis
_normalizers
))
1108 Qs
= self
.multiplication_table_from_matrix_basis(basis
)
1110 super(MatrixEuclideanJordanAlgebra
, self
).__init
__(field
,
1112 natural_basis
=basis
,
1117 def _charpoly_coefficients(self
):
1119 Override the parent method with something that tries to compute
1120 over a faster (non-extension) field.
1122 if self
._basis
_normalizers
is None:
1123 # We didn't normalize, so assume that the basis we started
1124 # with had entries in a nice field.
1125 return super(MatrixEuclideanJordanAlgebra
, self
)._charpoly
_coefficients
()
1127 basis
= ( (b
/n
) for (b
,n
) in zip(self
.natural_basis(),
1128 self
._basis
_normalizers
) )
1130 # Do this over the rationals and convert back at the end.
1131 # Only works because we know the entries of the basis are
1132 # integers. The argument ``check_axioms=False`` is required
1133 # because the trace inner-product method for this
1134 # class is a stub and can't actually be checked.
1135 J
= MatrixEuclideanJordanAlgebra(QQ
,
1137 normalize_basis
=False,
1140 a
= J
._charpoly
_coefficients
()
1142 # Unfortunately, changing the basis does change the
1143 # coefficients of the characteristic polynomial, but since
1144 # these are really the coefficients of the "characteristic
1145 # polynomial of" function, everything is still nice and
1146 # unevaluated. It's therefore "obvious" how scaling the
1147 # basis affects the coordinate variables X1, X2, et
1148 # cetera. Scaling the first basis vector up by "n" adds a
1149 # factor of 1/n into every "X1" term, for example. So here
1150 # we simply undo the basis_normalizer scaling that we
1151 # performed earlier.
1153 # The a[0] access here is safe because trivial algebras
1154 # won't have any basis normalizers and therefore won't
1155 # make it to this "else" branch.
1156 XS
= a
[0].parent().gens()
1157 subs_dict
= { XS
[i
]: self
._basis
_normalizers
[i
]*XS
[i
]
1158 for i
in range(len(XS
)) }
1159 return tuple( a_i
.subs(subs_dict
) for a_i
in a
)
1163 def multiplication_table_from_matrix_basis(basis
):
1165 At least three of the five simple Euclidean Jordan algebras have the
1166 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1167 multiplication on the right is matrix multiplication. Given a basis
1168 for the underlying matrix space, this function returns a
1169 multiplication table (obtained by looping through the basis
1170 elements) for an algebra of those matrices.
1172 # In S^2, for example, we nominally have four coordinates even
1173 # though the space is of dimension three only. The vector space V
1174 # is supposed to hold the entire long vector, and the subspace W
1175 # of V will be spanned by the vectors that arise from symmetric
1176 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1180 field
= basis
[0].base_ring()
1181 dimension
= basis
[0].nrows()
1183 V
= VectorSpace(field
, dimension
**2)
1184 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
1186 mult_table
= [[W
.zero() for j
in range(n
)] for i
in range(n
)]
1189 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
1190 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
1198 Embed the matrix ``M`` into a space of real matrices.
1200 The matrix ``M`` can have entries in any field at the moment:
1201 the real numbers, complex numbers, or quaternions. And although
1202 they are not a field, we can probably support octonions at some
1203 point, too. This function returns a real matrix that "acts like"
1204 the original with respect to matrix multiplication; i.e.
1206 real_embed(M*N) = real_embed(M)*real_embed(N)
1209 raise NotImplementedError
1213 def real_unembed(M
):
1215 The inverse of :meth:`real_embed`.
1217 raise NotImplementedError
1221 def natural_inner_product(cls
,X
,Y
):
1222 Xu
= cls
.real_unembed(X
)
1223 Yu
= cls
.real_unembed(Y
)
1224 tr
= (Xu
*Yu
).trace()
1227 # Works in QQ, AA, RDF, et cetera.
1229 except AttributeError:
1230 # A quaternion doesn't have a real() method, but does
1231 # have coefficient_tuple() method that returns the
1232 # coefficients of 1, i, j, and k -- in that order.
1233 return tr
.coefficient_tuple()[0]
1236 class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1240 The identity function, for embedding real matrices into real
1246 def real_unembed(M
):
1248 The identity function, for unembedding real matrices from real
1254 class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra
):
1256 The rank-n simple EJA consisting of real symmetric n-by-n
1257 matrices, the usual symmetric Jordan product, and the trace inner
1258 product. It has dimension `(n^2 + n)/2` over the reals.
1262 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1266 sage: J = RealSymmetricEJA(2)
1267 sage: e0, e1, e2 = J.gens()
1275 In theory, our "field" can be any subfield of the reals::
1277 sage: RealSymmetricEJA(2, RDF)
1278 Euclidean Jordan algebra of dimension 3 over Real Double Field
1279 sage: RealSymmetricEJA(2, RR)
1280 Euclidean Jordan algebra of dimension 3 over Real Field with
1281 53 bits of precision
1285 The dimension of this algebra is `(n^2 + n) / 2`::
1287 sage: set_random_seed()
1288 sage: n_max = RealSymmetricEJA._max_test_case_size()
1289 sage: n = ZZ.random_element(1, n_max)
1290 sage: J = RealSymmetricEJA(n)
1291 sage: J.dimension() == (n^2 + n)/2
1294 The Jordan multiplication is what we think it is::
1296 sage: set_random_seed()
1297 sage: J = RealSymmetricEJA.random_instance()
1298 sage: x,y = J.random_elements(2)
1299 sage: actual = (x*y).natural_representation()
1300 sage: X = x.natural_representation()
1301 sage: Y = y.natural_representation()
1302 sage: expected = (X*Y + Y*X)/2
1303 sage: actual == expected
1305 sage: J(expected) == x*y
1308 We can change the generator prefix::
1310 sage: RealSymmetricEJA(3, prefix='q').gens()
1311 (q0, q1, q2, q3, q4, q5)
1313 Our natural basis is normalized with respect to the natural inner
1314 product unless we specify otherwise::
1316 sage: set_random_seed()
1317 sage: J = RealSymmetricEJA.random_instance()
1318 sage: all( b.norm() == 1 for b in J.gens() )
1321 Since our natural basis is normalized with respect to the natural
1322 inner product, and since we know that this algebra is an EJA, any
1323 left-multiplication operator's matrix will be symmetric because
1324 natural->EJA basis representation is an isometry and within the EJA
1325 the operator is self-adjoint by the Jordan axiom::
1327 sage: set_random_seed()
1328 sage: x = RealSymmetricEJA.random_instance().random_element()
1329 sage: x.operator().matrix().is_symmetric()
1332 We can construct the (trivial) algebra of rank zero::
1334 sage: RealSymmetricEJA(0)
1335 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
1339 def _denormalized_basis(cls
, n
, field
):
1341 Return a basis for the space of real symmetric n-by-n matrices.
1345 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1349 sage: set_random_seed()
1350 sage: n = ZZ.random_element(1,5)
1351 sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
1352 sage: all( M.is_symmetric() for M in B)
1356 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1360 for j
in range(i
+1):
1361 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1365 Sij
= Eij
+ Eij
.transpose()
1371 def _max_test_case_size():
1372 return 4 # Dimension 10
1375 def __init__(self
, n
, field
=AA
, **kwargs
):
1376 basis
= self
._denormalized
_basis
(n
, field
)
1377 super(RealSymmetricEJA
, self
).__init
__(field
,
1381 self
.rank
.set_cache(n
)
1384 class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1388 Embed the n-by-n complex matrix ``M`` into the space of real
1389 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1390 bi` to the block matrix ``[[a,b],[-b,a]]``.
1394 sage: from mjo.eja.eja_algebra import \
1395 ....: ComplexMatrixEuclideanJordanAlgebra
1399 sage: F = QuadraticField(-1, 'I')
1400 sage: x1 = F(4 - 2*i)
1401 sage: x2 = F(1 + 2*i)
1404 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1405 sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1414 Embedding is a homomorphism (isomorphism, in fact)::
1416 sage: set_random_seed()
1417 sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
1418 sage: n = ZZ.random_element(n_max)
1419 sage: F = QuadraticField(-1, 'I')
1420 sage: X = random_matrix(F, n)
1421 sage: Y = random_matrix(F, n)
1422 sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
1423 sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
1424 sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1431 raise ValueError("the matrix 'M' must be square")
1433 # We don't need any adjoined elements...
1434 field
= M
.base_ring().base_ring()
1438 a
= z
.list()[0] # real part, I guess
1439 b
= z
.list()[1] # imag part, I guess
1440 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1442 return matrix
.block(field
, n
, blocks
)
1446 def real_unembed(M
):
1448 The inverse of _embed_complex_matrix().
1452 sage: from mjo.eja.eja_algebra import \
1453 ....: ComplexMatrixEuclideanJordanAlgebra
1457 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1458 ....: [-2, 1, -4, 3],
1459 ....: [ 9, 10, 11, 12],
1460 ....: [-10, 9, -12, 11] ])
1461 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
1463 [ 10*I + 9 12*I + 11]
1467 Unembedding is the inverse of embedding::
1469 sage: set_random_seed()
1470 sage: F = QuadraticField(-1, 'I')
1471 sage: M = random_matrix(F, 3)
1472 sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1473 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1479 raise ValueError("the matrix 'M' must be square")
1480 if not n
.mod(2).is_zero():
1481 raise ValueError("the matrix 'M' must be a complex embedding")
1483 # If "M" was normalized, its base ring might have roots
1484 # adjoined and they can stick around after unembedding.
1485 field
= M
.base_ring()
1486 R
= PolynomialRing(field
, 'z')
1489 # Sage doesn't know how to embed AA into QQbar, i.e. how
1490 # to adjoin sqrt(-1) to AA.
1493 F
= field
.extension(z
**2 + 1, 'I', embedding
=CLF(-1).sqrt())
1496 # Go top-left to bottom-right (reading order), converting every
1497 # 2-by-2 block we see to a single complex element.
1499 for k
in range(n
/2):
1500 for j
in range(n
/2):
1501 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1502 if submat
[0,0] != submat
[1,1]:
1503 raise ValueError('bad on-diagonal submatrix')
1504 if submat
[0,1] != -submat
[1,0]:
1505 raise ValueError('bad off-diagonal submatrix')
1506 z
= submat
[0,0] + submat
[0,1]*i
1509 return matrix(F
, n
/2, elements
)
1513 def natural_inner_product(cls
,X
,Y
):
1515 Compute a natural inner product in this algebra directly from
1520 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1524 This gives the same answer as the slow, default method implemented
1525 in :class:`MatrixEuclideanJordanAlgebra`::
1527 sage: set_random_seed()
1528 sage: J = ComplexHermitianEJA.random_instance()
1529 sage: x,y = J.random_elements(2)
1530 sage: Xe = x.natural_representation()
1531 sage: Ye = y.natural_representation()
1532 sage: X = ComplexHermitianEJA.real_unembed(Xe)
1533 sage: Y = ComplexHermitianEJA.real_unembed(Ye)
1534 sage: expected = (X*Y).trace().real()
1535 sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
1536 sage: actual == expected
1540 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/2
1543 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra
):
1545 The rank-n simple EJA consisting of complex Hermitian n-by-n
1546 matrices over the real numbers, the usual symmetric Jordan product,
1547 and the real-part-of-trace inner product. It has dimension `n^2` over
1552 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1556 In theory, our "field" can be any subfield of the reals::
1558 sage: ComplexHermitianEJA(2, RDF)
1559 Euclidean Jordan algebra of dimension 4 over Real Double Field
1560 sage: ComplexHermitianEJA(2, RR)
1561 Euclidean Jordan algebra of dimension 4 over Real Field with
1562 53 bits of precision
1566 The dimension of this algebra is `n^2`::
1568 sage: set_random_seed()
1569 sage: n_max = ComplexHermitianEJA._max_test_case_size()
1570 sage: n = ZZ.random_element(1, n_max)
1571 sage: J = ComplexHermitianEJA(n)
1572 sage: J.dimension() == n^2
1575 The Jordan multiplication is what we think it is::
1577 sage: set_random_seed()
1578 sage: J = ComplexHermitianEJA.random_instance()
1579 sage: x,y = J.random_elements(2)
1580 sage: actual = (x*y).natural_representation()
1581 sage: X = x.natural_representation()
1582 sage: Y = y.natural_representation()
1583 sage: expected = (X*Y + Y*X)/2
1584 sage: actual == expected
1586 sage: J(expected) == x*y
1589 We can change the generator prefix::
1591 sage: ComplexHermitianEJA(2, prefix='z').gens()
1594 Our natural basis is normalized with respect to the natural inner
1595 product unless we specify otherwise::
1597 sage: set_random_seed()
1598 sage: J = ComplexHermitianEJA.random_instance()
1599 sage: all( b.norm() == 1 for b in J.gens() )
1602 Since our natural basis is normalized with respect to the natural
1603 inner product, and since we know that this algebra is an EJA, any
1604 left-multiplication operator's matrix will be symmetric because
1605 natural->EJA basis representation is an isometry and within the EJA
1606 the operator is self-adjoint by the Jordan axiom::
1608 sage: set_random_seed()
1609 sage: x = ComplexHermitianEJA.random_instance().random_element()
1610 sage: x.operator().matrix().is_symmetric()
1613 We can construct the (trivial) algebra of rank zero::
1615 sage: ComplexHermitianEJA(0)
1616 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
1621 def _denormalized_basis(cls
, n
, field
):
1623 Returns a basis for the space of complex Hermitian n-by-n matrices.
1625 Why do we embed these? Basically, because all of numerical linear
1626 algebra assumes that you're working with vectors consisting of `n`
1627 entries from a field and scalars from the same field. There's no way
1628 to tell SageMath that (for example) the vectors contain complex
1629 numbers, while the scalar field is real.
1633 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1637 sage: set_random_seed()
1638 sage: n = ZZ.random_element(1,5)
1639 sage: field = QuadraticField(2, 'sqrt2')
1640 sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
1641 sage: all( M.is_symmetric() for M in B)
1645 R
= PolynomialRing(field
, 'z')
1647 F
= field
.extension(z
**2 + 1, 'I')
1650 # This is like the symmetric case, but we need to be careful:
1652 # * We want conjugate-symmetry, not just symmetry.
1653 # * The diagonal will (as a result) be real.
1657 for j
in range(i
+1):
1658 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1660 Sij
= cls
.real_embed(Eij
)
1663 # The second one has a minus because it's conjugated.
1664 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1666 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1669 # Since we embedded these, we can drop back to the "field" that we
1670 # started with instead of the complex extension "F".
1671 return ( s
.change_ring(field
) for s
in S
)
1674 def __init__(self
, n
, field
=AA
, **kwargs
):
1675 basis
= self
._denormalized
_basis
(n
,field
)
1676 super(ComplexHermitianEJA
,self
).__init
__(field
,
1680 self
.rank
.set_cache(n
)
1683 class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1687 Embed the n-by-n quaternion matrix ``M`` into the space of real
1688 matrices of size 4n-by-4n by first sending each quaternion entry `z
1689 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1690 c+di],[-c + di, a-bi]]`, and then embedding those into a real
1695 sage: from mjo.eja.eja_algebra import \
1696 ....: QuaternionMatrixEuclideanJordanAlgebra
1700 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1701 sage: i,j,k = Q.gens()
1702 sage: x = 1 + 2*i + 3*j + 4*k
1703 sage: M = matrix(Q, 1, [[x]])
1704 sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1710 Embedding is a homomorphism (isomorphism, in fact)::
1712 sage: set_random_seed()
1713 sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
1714 sage: n = ZZ.random_element(n_max)
1715 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1716 sage: X = random_matrix(Q, n)
1717 sage: Y = random_matrix(Q, n)
1718 sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
1719 sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
1720 sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1725 quaternions
= M
.base_ring()
1728 raise ValueError("the matrix 'M' must be square")
1730 F
= QuadraticField(-1, 'I')
1735 t
= z
.coefficient_tuple()
1740 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1741 [-c
+ d
*i
, a
- b
*i
]])
1742 realM
= ComplexMatrixEuclideanJordanAlgebra
.real_embed(cplxM
)
1743 blocks
.append(realM
)
1745 # We should have real entries by now, so use the realest field
1746 # we've got for the return value.
1747 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1752 def real_unembed(M
):
1754 The inverse of _embed_quaternion_matrix().
1758 sage: from mjo.eja.eja_algebra import \
1759 ....: QuaternionMatrixEuclideanJordanAlgebra
1763 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1764 ....: [-2, 1, -4, 3],
1765 ....: [-3, 4, 1, -2],
1766 ....: [-4, -3, 2, 1]])
1767 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
1768 [1 + 2*i + 3*j + 4*k]
1772 Unembedding is the inverse of embedding::
1774 sage: set_random_seed()
1775 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1776 sage: M = random_matrix(Q, 3)
1777 sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1778 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1784 raise ValueError("the matrix 'M' must be square")
1785 if not n
.mod(4).is_zero():
1786 raise ValueError("the matrix 'M' must be a quaternion embedding")
1788 # Use the base ring of the matrix to ensure that its entries can be
1789 # multiplied by elements of the quaternion algebra.
1790 field
= M
.base_ring()
1791 Q
= QuaternionAlgebra(field
,-1,-1)
1794 # Go top-left to bottom-right (reading order), converting every
1795 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1798 for l
in range(n
/4):
1799 for m
in range(n
/4):
1800 submat
= ComplexMatrixEuclideanJordanAlgebra
.real_unembed(
1801 M
[4*l
:4*l
+4,4*m
:4*m
+4] )
1802 if submat
[0,0] != submat
[1,1].conjugate():
1803 raise ValueError('bad on-diagonal submatrix')
1804 if submat
[0,1] != -submat
[1,0].conjugate():
1805 raise ValueError('bad off-diagonal submatrix')
1806 z
= submat
[0,0].real()
1807 z
+= submat
[0,0].imag()*i
1808 z
+= submat
[0,1].real()*j
1809 z
+= submat
[0,1].imag()*k
1812 return matrix(Q
, n
/4, elements
)
1816 def natural_inner_product(cls
,X
,Y
):
1818 Compute a natural inner product in this algebra directly from
1823 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1827 This gives the same answer as the slow, default method implemented
1828 in :class:`MatrixEuclideanJordanAlgebra`::
1830 sage: set_random_seed()
1831 sage: J = QuaternionHermitianEJA.random_instance()
1832 sage: x,y = J.random_elements(2)
1833 sage: Xe = x.natural_representation()
1834 sage: Ye = y.natural_representation()
1835 sage: X = QuaternionHermitianEJA.real_unembed(Xe)
1836 sage: Y = QuaternionHermitianEJA.real_unembed(Ye)
1837 sage: expected = (X*Y).trace().coefficient_tuple()[0]
1838 sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye)
1839 sage: actual == expected
1843 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/4
1846 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra
):
1848 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1849 matrices, the usual symmetric Jordan product, and the
1850 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1855 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1859 In theory, our "field" can be any subfield of the reals::
1861 sage: QuaternionHermitianEJA(2, RDF)
1862 Euclidean Jordan algebra of dimension 6 over Real Double Field
1863 sage: QuaternionHermitianEJA(2, RR)
1864 Euclidean Jordan algebra of dimension 6 over Real Field with
1865 53 bits of precision
1869 The dimension of this algebra is `2*n^2 - n`::
1871 sage: set_random_seed()
1872 sage: n_max = QuaternionHermitianEJA._max_test_case_size()
1873 sage: n = ZZ.random_element(1, n_max)
1874 sage: J = QuaternionHermitianEJA(n)
1875 sage: J.dimension() == 2*(n^2) - n
1878 The Jordan multiplication is what we think it is::
1880 sage: set_random_seed()
1881 sage: J = QuaternionHermitianEJA.random_instance()
1882 sage: x,y = J.random_elements(2)
1883 sage: actual = (x*y).natural_representation()
1884 sage: X = x.natural_representation()
1885 sage: Y = y.natural_representation()
1886 sage: expected = (X*Y + Y*X)/2
1887 sage: actual == expected
1889 sage: J(expected) == x*y
1892 We can change the generator prefix::
1894 sage: QuaternionHermitianEJA(2, prefix='a').gens()
1895 (a0, a1, a2, a3, a4, a5)
1897 Our natural basis is normalized with respect to the natural inner
1898 product unless we specify otherwise::
1900 sage: set_random_seed()
1901 sage: J = QuaternionHermitianEJA.random_instance()
1902 sage: all( b.norm() == 1 for b in J.gens() )
1905 Since our natural basis is normalized with respect to the natural
1906 inner product, and since we know that this algebra is an EJA, any
1907 left-multiplication operator's matrix will be symmetric because
1908 natural->EJA basis representation is an isometry and within the EJA
1909 the operator is self-adjoint by the Jordan axiom::
1911 sage: set_random_seed()
1912 sage: x = QuaternionHermitianEJA.random_instance().random_element()
1913 sage: x.operator().matrix().is_symmetric()
1916 We can construct the (trivial) algebra of rank zero::
1918 sage: QuaternionHermitianEJA(0)
1919 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
1923 def _denormalized_basis(cls
, n
, field
):
1925 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1927 Why do we embed these? Basically, because all of numerical
1928 linear algebra assumes that you're working with vectors consisting
1929 of `n` entries from a field and scalars from the same field. There's
1930 no way to tell SageMath that (for example) the vectors contain
1931 complex numbers, while the scalar field is real.
1935 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1939 sage: set_random_seed()
1940 sage: n = ZZ.random_element(1,5)
1941 sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
1942 sage: all( M.is_symmetric() for M in B )
1946 Q
= QuaternionAlgebra(QQ
,-1,-1)
1949 # This is like the symmetric case, but we need to be careful:
1951 # * We want conjugate-symmetry, not just symmetry.
1952 # * The diagonal will (as a result) be real.
1956 for j
in range(i
+1):
1957 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1959 Sij
= cls
.real_embed(Eij
)
1962 # The second, third, and fourth ones have a minus
1963 # because they're conjugated.
1964 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1966 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1968 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
1970 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
1973 # Since we embedded these, we can drop back to the "field" that we
1974 # started with instead of the quaternion algebra "Q".
1975 return ( s
.change_ring(field
) for s
in S
)
1978 def __init__(self
, n
, field
=AA
, **kwargs
):
1979 basis
= self
._denormalized
_basis
(n
,field
)
1980 super(QuaternionHermitianEJA
,self
).__init
__(field
,
1984 self
.rank
.set_cache(n
)
1987 class HadamardEJA(RationalBasisEuclideanJordanAlgebra
):
1989 Return the Euclidean Jordan Algebra corresponding to the set
1990 `R^n` under the Hadamard product.
1992 Note: this is nothing more than the Cartesian product of ``n``
1993 copies of the spin algebra. Once Cartesian product algebras
1994 are implemented, this can go.
1998 sage: from mjo.eja.eja_algebra import HadamardEJA
2002 This multiplication table can be verified by hand::
2004 sage: J = HadamardEJA(3)
2005 sage: e0,e1,e2 = J.gens()
2021 We can change the generator prefix::
2023 sage: HadamardEJA(3, prefix='r').gens()
2027 def __init__(self
, n
, field
=AA
, **kwargs
):
2028 V
= VectorSpace(field
, n
)
2029 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in range(n
) ]
2032 super(HadamardEJA
, self
).__init
__(field
,
2036 self
.rank
.set_cache(n
)
2038 def inner_product(self
, x
, y
):
2040 Faster to reimplement than to use natural representations.
2044 sage: from mjo.eja.eja_algebra import HadamardEJA
2048 Ensure that this is the usual inner product for the algebras
2051 sage: set_random_seed()
2052 sage: J = HadamardEJA.random_instance()
2053 sage: x,y = J.random_elements(2)
2054 sage: X = x.natural_representation()
2055 sage: Y = y.natural_representation()
2056 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
2060 return x
.to_vector().inner_product(y
.to_vector())
2063 class BilinearFormEJA(RationalBasisEuclideanJordanAlgebra
):
2065 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2066 with the half-trace inner product and jordan product ``x*y =
2067 (x0*y0 + <B*x_bar,y_bar>, x0*y_bar + y0*x_bar)`` where ``B`` is a
2068 symmetric positive-definite "bilinear form" matrix. It has
2069 dimension `n` over the reals, and reduces to the ``JordanSpinEJA``
2070 when ``B`` is the identity matrix of order ``n-1``.
2074 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
2075 ....: JordanSpinEJA)
2079 When no bilinear form is specified, the identity matrix is used,
2080 and the resulting algebra is the Jordan spin algebra::
2082 sage: J0 = BilinearFormEJA(3)
2083 sage: J1 = JordanSpinEJA(3)
2084 sage: J0.multiplication_table() == J0.multiplication_table()
2089 We can create a zero-dimensional algebra::
2091 sage: J = BilinearFormEJA(0)
2095 We can check the multiplication condition given in the Jordan, von
2096 Neumann, and Wigner paper (and also discussed on my "On the
2097 symmetry..." paper). Note that this relies heavily on the standard
2098 choice of basis, as does anything utilizing the bilinear form matrix::
2100 sage: set_random_seed()
2101 sage: n = ZZ.random_element(5)
2102 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
2103 sage: B = M.transpose()*M
2104 sage: J = BilinearFormEJA(n, B=B)
2105 sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
2106 sage: V = J.vector_space()
2107 sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
2108 ....: for ei in eis ]
2109 sage: actual = [ sis[i]*sis[j]
2110 ....: for i in range(n-1)
2111 ....: for j in range(n-1) ]
2112 sage: expected = [ J.one() if i == j else J.zero()
2113 ....: for i in range(n-1)
2114 ....: for j in range(n-1) ]
2115 sage: actual == expected
2118 def __init__(self
, n
, field
=AA
, B
=None, **kwargs
):
2120 self
._B
= matrix
.identity(field
, max(0,n
-1))
2124 V
= VectorSpace(field
, n
)
2125 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
2134 z0
= x0
*y0
+ (self
._B
*xbar
).inner_product(ybar
)
2135 zbar
= y0
*xbar
+ x0
*ybar
2136 z
= V([z0
] + zbar
.list())
2137 mult_table
[i
][j
] = z
2139 # The rank of this algebra is two, unless we're in a
2140 # one-dimensional ambient space (because the rank is bounded
2141 # by the ambient dimension).
2142 super(BilinearFormEJA
, self
).__init
__(field
,
2146 self
.rank
.set_cache(min(n
,2))
2148 def inner_product(self
, x
, y
):
2150 Half of the trace inner product.
2152 This is defined so that the special case of the Jordan spin
2153 algebra gets the usual inner product.
2157 sage: from mjo.eja.eja_algebra import BilinearFormEJA
2161 Ensure that this is one-half of the trace inner-product when
2162 the algebra isn't just the reals (when ``n`` isn't one). This
2163 is in Faraut and Koranyi, and also my "On the symmetry..."
2166 sage: set_random_seed()
2167 sage: n = ZZ.random_element(2,5)
2168 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
2169 sage: B = M.transpose()*M
2170 sage: J = BilinearFormEJA(n, B=B)
2171 sage: x = J.random_element()
2172 sage: y = J.random_element()
2173 sage: x.inner_product(y) == (x*y).trace()/2
2177 xvec
= x
.to_vector()
2179 yvec
= y
.to_vector()
2181 return x
[0]*y
[0] + (self
._B
*xbar
).inner_product(ybar
)
2184 class JordanSpinEJA(BilinearFormEJA
):
2186 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2187 with the usual inner product and jordan product ``x*y =
2188 (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2193 sage: from mjo.eja.eja_algebra import JordanSpinEJA
2197 This multiplication table can be verified by hand::
2199 sage: J = JordanSpinEJA(4)
2200 sage: e0,e1,e2,e3 = J.gens()
2216 We can change the generator prefix::
2218 sage: JordanSpinEJA(2, prefix='B').gens()
2223 Ensure that we have the usual inner product on `R^n`::
2225 sage: set_random_seed()
2226 sage: J = JordanSpinEJA.random_instance()
2227 sage: x,y = J.random_elements(2)
2228 sage: X = x.natural_representation()
2229 sage: Y = y.natural_representation()
2230 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
2234 def __init__(self
, n
, field
=AA
, **kwargs
):
2235 # This is a special case of the BilinearFormEJA with the identity
2236 # matrix as its bilinear form.
2237 super(JordanSpinEJA
, self
).__init
__(n
, field
, **kwargs
)
2240 class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2242 The trivial Euclidean Jordan algebra consisting of only a zero element.
2246 sage: from mjo.eja.eja_algebra import TrivialEJA
2250 sage: J = TrivialEJA()
2257 sage: 7*J.one()*12*J.one()
2259 sage: J.one().inner_product(J.one())
2261 sage: J.one().norm()
2263 sage: J.one().subalgebra_generated_by()
2264 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
2269 def __init__(self
, field
=AA
, **kwargs
):
2271 super(TrivialEJA
, self
).__init
__(field
,
2275 # The rank is zero using my definition, namely the dimension of the
2276 # largest subalgebra generated by any element.
2277 self
.rank
.set_cache(0)
2280 class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2282 The external (orthogonal) direct sum of two other Euclidean Jordan
2283 algebras. Essentially the Cartesian product of its two factors.
2284 Every Euclidean Jordan algebra decomposes into an orthogonal
2285 direct sum of simple Euclidean Jordan algebras, so no generality
2286 is lost by providing only this construction.
2290 sage: from mjo.eja.eja_algebra import (HadamardEJA,
2291 ....: RealSymmetricEJA,
2296 sage: J1 = HadamardEJA(2)
2297 sage: J2 = RealSymmetricEJA(3)
2298 sage: J = DirectSumEJA(J1,J2)
2305 def __init__(self
, J1
, J2
, field
=AA
, **kwargs
):
2309 V
= VectorSpace(field
, n
)
2310 mult_table
= [ [ V
.zero() for j
in range(n
) ]
2314 p
= (J1
.monomial(i
)*J1
.monomial(j
)).to_vector()
2315 mult_table
[i
][j
] = V(p
.list() + [field
.zero()]*n2
)
2319 p
= (J2
.monomial(i
)*J2
.monomial(j
)).to_vector()
2320 mult_table
[n1
+i
][n1
+j
] = V([field
.zero()]*n1
+ p
.list())
2322 super(DirectSumEJA
, self
).__init
__(field
,
2326 self
.rank
.set_cache(J1
.rank() + J2
.rank())