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.
10 sage: from mjo.eja.eja_algebra import random_eja
15 Euclidean Jordan algebra of dimension...
19 from itertools
import repeat
21 from sage
.algebras
.quatalg
.quaternion_algebra
import QuaternionAlgebra
22 from sage
.categories
.magmatic_algebras
import MagmaticAlgebras
23 from sage
.combinat
.free_module
import CombinatorialFreeModule
24 from sage
.matrix
.constructor
import matrix
25 from sage
.matrix
.matrix_space
import MatrixSpace
26 from sage
.misc
.cachefunc
import cached_method
27 from sage
.misc
.table
import table
28 from sage
.modules
.free_module
import FreeModule
, VectorSpace
29 from sage
.rings
.all
import (ZZ
, QQ
, AA
, QQbar
, RR
, RLF
, CLF
,
32 from mjo
.eja
.eja_element
import FiniteDimensionalEJAElement
33 from mjo
.eja
.eja_operator
import FiniteDimensionalEJAOperator
34 from mjo
.eja
.eja_utils
import _mat2vec
36 class FiniteDimensionalEJA(CombinatorialFreeModule
):
38 A finite-dimensional Euclidean Jordan algebra.
42 - basis -- a tuple of basis elements in their matrix form.
44 - jordan_product -- function of two elements (in matrix form)
45 that returns their jordan product in this algebra; this will
46 be applied to ``basis`` to compute a multiplication table for
49 - inner_product -- function of two elements (in matrix form) that
50 returns their inner product. This will be applied to ``basis`` to
51 compute an inner-product table (basically a matrix) for this algebra.
54 Element
= FiniteDimensionalEJAElement
68 if not field
.is_subring(RR
):
69 # Note: this does return true for the real algebraic
70 # field, the rationals, and any quadratic field where
71 # we've specified a real embedding.
72 raise ValueError("scalar field is not real")
74 # If the basis given to us wasn't over the field that it's
75 # supposed to be over, fix that. Or, you know, crash.
76 basis
= tuple( b
.change_ring(field
) for b
in basis
)
79 # Check commutativity of the Jordan and inner-products.
80 # This has to be done before we build the multiplication
81 # and inner-product tables/matrices, because we take
82 # advantage of symmetry in the process.
83 if not all( jordan_product(bi
,bj
) == jordan_product(bj
,bi
)
86 raise ValueError("Jordan product is not commutative")
88 if not all( inner_product(bi
,bj
) == inner_product(bj
,bi
)
91 raise ValueError("inner-product is not commutative")
94 category
= MagmaticAlgebras(field
).FiniteDimensional()
95 category
= category
.WithBasis().Unital()
97 # Element subalgebras can take advantage of this.
98 category
= category
.Associative()
100 # Call the superclass constructor so that we can use its from_vector()
101 # method to build our multiplication table.
103 super().__init
__(field
,
109 # Now comes all of the hard work. We'll be constructing an
110 # ambient vector space V that our (vectorized) basis lives in,
111 # as well as a subspace W of V spanned by those (vectorized)
112 # basis elements. The W-coordinates are the coefficients that
113 # we see in things like x = 1*e1 + 2*e2.
118 # Works on both column and square matrices...
119 degree
= len(basis
[0].list())
121 # Build an ambient space that fits our matrix basis when
122 # written out as "long vectors."
123 V
= VectorSpace(field
, degree
)
125 # The matrix that will hole the orthonormal -> unorthonormal
126 # coordinate transformation.
127 self
._deortho
_matrix
= None
130 # Save a copy of the un-orthonormalized basis for later.
131 # Convert it to ambient V (vector) coordinates while we're
132 # at it, because we'd have to do it later anyway.
133 deortho_vector_basis
= tuple( V(b
.list()) for b
in basis
)
135 from mjo
.eja
.eja_utils
import gram_schmidt
136 basis
= gram_schmidt(basis
, inner_product
)
138 # Save the (possibly orthonormalized) matrix basis for
140 self
._matrix
_basis
= basis
142 # Now create the vector space for the algebra, which will have
143 # its own set of non-ambient coordinates (in terms of the
145 vector_basis
= tuple( V(b
.list()) for b
in basis
)
146 W
= V
.span_of_basis( vector_basis
, check
=check_axioms
)
149 # Now "W" is the vector space of our algebra coordinates. The
150 # variables "X1", "X2",... refer to the entries of vectors in
151 # W. Thus to convert back and forth between the orthonormal
152 # coordinates and the given ones, we need to stick the original
154 U
= V
.span_of_basis( deortho_vector_basis
, check
=check_axioms
)
155 self
._deortho
_matrix
= matrix( U
.coordinate_vector(q
)
156 for q
in vector_basis
)
159 # Now we actually compute the multiplication and inner-product
160 # tables/matrices using the possibly-orthonormalized basis.
161 self
._inner
_product
_matrix
= matrix
.identity(field
, n
)
162 self
._multiplication
_table
= [ [0 for j
in range(i
+1)]
165 # Note: the Jordan and inner-products are defined in terms
166 # of the ambient basis. It's important that their arguments
167 # are in ambient coordinates as well.
170 # ortho basis w.r.t. ambient coords
174 # The jordan product returns a matrixy answer, so we
175 # have to convert it to the algebra coordinates.
176 elt
= jordan_product(q_i
, q_j
)
177 elt
= W
.coordinate_vector(V(elt
.list()))
178 self
._multiplication
_table
[i
][j
] = self
.from_vector(elt
)
180 if not orthonormalize
:
181 # If we're orthonormalizing the basis with respect
182 # to an inner-product, then the inner-product
183 # matrix with respect to the resulting basis is
184 # just going to be the identity.
185 ip
= inner_product(q_i
, q_j
)
186 self
._inner
_product
_matrix
[i
,j
] = ip
187 self
._inner
_product
_matrix
[j
,i
] = ip
189 self
._inner
_product
_matrix
._cache
= {'hermitian': True}
190 self
._inner
_product
_matrix
.set_immutable()
193 if not self
._is
_jordanian
():
194 raise ValueError("Jordan identity does not hold")
195 if not self
._inner
_product
_is
_associative
():
196 raise ValueError("inner product is not associative")
199 def _coerce_map_from_base_ring(self
):
201 Disable the map from the base ring into the algebra.
203 Performing a nonsense conversion like this automatically
204 is counterpedagogical. The fallback is to try the usual
205 element constructor, which should also fail.
209 sage: from mjo.eja.eja_algebra import random_eja
213 sage: set_random_seed()
214 sage: J = random_eja()
216 Traceback (most recent call last):
218 ValueError: not an element of this algebra
224 def product_on_basis(self
, i
, j
):
225 # We only stored the lower-triangular portion of the
226 # multiplication table.
228 return self
._multiplication
_table
[i
][j
]
230 return self
._multiplication
_table
[j
][i
]
232 def inner_product(self
, x
, y
):
234 The inner product associated with this Euclidean Jordan algebra.
236 Defaults to the trace inner product, but can be overridden by
237 subclasses if they are sure that the necessary properties are
242 sage: from mjo.eja.eja_algebra import (random_eja,
244 ....: BilinearFormEJA)
248 Our inner product is "associative," which means the following for
249 a symmetric bilinear form::
251 sage: set_random_seed()
252 sage: J = random_eja()
253 sage: x,y,z = J.random_elements(3)
254 sage: (x*y).inner_product(z) == y.inner_product(x*z)
259 Ensure that this is the usual inner product for the algebras
262 sage: set_random_seed()
263 sage: J = HadamardEJA.random_instance()
264 sage: x,y = J.random_elements(2)
265 sage: actual = x.inner_product(y)
266 sage: expected = x.to_vector().inner_product(y.to_vector())
267 sage: actual == expected
270 Ensure that this is one-half of the trace inner-product in a
271 BilinearFormEJA that isn't just the reals (when ``n`` isn't
272 one). This is in Faraut and Koranyi, and also my "On the
275 sage: set_random_seed()
276 sage: J = BilinearFormEJA.random_instance()
277 sage: n = J.dimension()
278 sage: x = J.random_element()
279 sage: y = J.random_element()
280 sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2)
283 B
= self
._inner
_product
_matrix
284 return (B
*x
.to_vector()).inner_product(y
.to_vector())
287 def _is_commutative(self
):
289 Whether or not this algebra's multiplication table is commutative.
291 This method should of course always return ``True``, unless
292 this algebra was constructed with ``check_axioms=False`` and
293 passed an invalid multiplication table.
295 return all( self
.product_on_basis(i
,j
) == self
.product_on_basis(i
,j
)
296 for i
in range(self
.dimension())
297 for j
in range(self
.dimension()) )
299 def _is_jordanian(self
):
301 Whether or not this algebra's multiplication table respects the
302 Jordan identity `(x^{2})(xy) = x(x^{2}y)`.
304 We only check one arrangement of `x` and `y`, so for a
305 ``True`` result to be truly true, you should also check
306 :meth:`_is_commutative`. This method should of course always
307 return ``True``, unless this algebra was constructed with
308 ``check_axioms=False`` and passed an invalid multiplication table.
310 return all( (self
.monomial(i
)**2)*(self
.monomial(i
)*self
.monomial(j
))
312 (self
.monomial(i
))*((self
.monomial(i
)**2)*self
.monomial(j
))
313 for i
in range(self
.dimension())
314 for j
in range(self
.dimension()) )
316 def _inner_product_is_associative(self
):
318 Return whether or not this algebra's inner product `B` is
319 associative; that is, whether or not `B(xy,z) = B(x,yz)`.
321 This method should of course always return ``True``, unless
322 this algebra was constructed with ``check_axioms=False`` and
323 passed an invalid Jordan or inner-product.
326 # Used to check whether or not something is zero in an inexact
327 # ring. This number is sufficient to allow the construction of
328 # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True.
331 for i
in range(self
.dimension()):
332 for j
in range(self
.dimension()):
333 for k
in range(self
.dimension()):
337 diff
= (x
*y
).inner_product(z
) - x
.inner_product(y
*z
)
339 if self
.base_ring().is_exact():
343 if diff
.abs() > epsilon
:
348 def _element_constructor_(self
, elt
):
350 Construct an element of this algebra from its vector or matrix
353 This gets called only after the parent element _call_ method
354 fails to find a coercion for the argument.
358 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
360 ....: RealSymmetricEJA)
364 The identity in `S^n` is converted to the identity in the EJA::
366 sage: J = RealSymmetricEJA(3)
367 sage: I = matrix.identity(QQ,3)
368 sage: J(I) == J.one()
371 This skew-symmetric matrix can't be represented in the EJA::
373 sage: J = RealSymmetricEJA(3)
374 sage: A = matrix(QQ,3, lambda i,j: i-j)
376 Traceback (most recent call last):
378 ValueError: not an element of this algebra
382 Ensure that we can convert any element of the two non-matrix
383 simple algebras (whose matrix representations are columns)
384 back and forth faithfully::
386 sage: set_random_seed()
387 sage: J = HadamardEJA.random_instance()
388 sage: x = J.random_element()
389 sage: J(x.to_vector().column()) == x
391 sage: J = JordanSpinEJA.random_instance()
392 sage: x = J.random_element()
393 sage: J(x.to_vector().column()) == x
397 msg
= "not an element of this algebra"
399 # The superclass implementation of random_element()
400 # needs to be able to coerce "0" into the algebra.
402 elif elt
in self
.base_ring():
403 # Ensure that no base ring -> algebra coercion is performed
404 # by this method. There's some stupidity in sage that would
405 # otherwise propagate to this method; for example, sage thinks
406 # that the integer 3 belongs to the space of 2-by-2 matrices.
407 raise ValueError(msg
)
411 except (AttributeError, TypeError):
412 # Try to convert a vector into a column-matrix
415 if elt
not in self
.matrix_space():
416 raise ValueError(msg
)
418 # Thanks for nothing! Matrix spaces aren't vector spaces in
419 # Sage, so we have to figure out its matrix-basis coordinates
420 # ourselves. We use the basis space's ring instead of the
421 # element's ring because the basis space might be an algebraic
422 # closure whereas the base ring of the 3-by-3 identity matrix
423 # could be QQ instead of QQbar.
425 # We pass check=False because the matrix basis is "guaranteed"
426 # to be linearly independent... right? Ha ha.
427 V
= VectorSpace(self
.base_ring(), elt
.nrows()*elt
.ncols())
428 W
= V
.span_of_basis( (_mat2vec(s
) for s
in self
.matrix_basis()),
432 coords
= W
.coordinate_vector(_mat2vec(elt
))
433 except ArithmeticError: # vector is not in free module
434 raise ValueError(msg
)
436 return self
.from_vector(coords
)
440 Return a string representation of ``self``.
444 sage: from mjo.eja.eja_algebra import JordanSpinEJA
448 Ensure that it says what we think it says::
450 sage: JordanSpinEJA(2, field=AA)
451 Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
452 sage: JordanSpinEJA(3, field=RDF)
453 Euclidean Jordan algebra of dimension 3 over Real Double Field
456 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
457 return fmt
.format(self
.dimension(), self
.base_ring())
461 def characteristic_polynomial_of(self
):
463 Return the algebra's "characteristic polynomial of" function,
464 which is itself a multivariate polynomial that, when evaluated
465 at the coordinates of some algebra element, returns that
466 element's characteristic polynomial.
468 The resulting polynomial has `n+1` variables, where `n` is the
469 dimension of this algebra. The first `n` variables correspond to
470 the coordinates of an algebra element: when evaluated at the
471 coordinates of an algebra element with respect to a certain
472 basis, the result is a univariate polynomial (in the one
473 remaining variable ``t``), namely the characteristic polynomial
478 sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA
482 The characteristic polynomial in the spin algebra is given in
483 Alizadeh, Example 11.11::
485 sage: J = JordanSpinEJA(3)
486 sage: p = J.characteristic_polynomial_of(); p
487 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
488 sage: xvec = J.one().to_vector()
492 By definition, the characteristic polynomial is a monic
493 degree-zero polynomial in a rank-zero algebra. Note that
494 Cayley-Hamilton is indeed satisfied since the polynomial
495 ``1`` evaluates to the identity element of the algebra on
498 sage: J = TrivialEJA()
499 sage: J.characteristic_polynomial_of()
506 # The list of coefficient polynomials a_0, a_1, a_2, ..., a_(r-1).
507 a
= self
._charpoly
_coefficients
()
509 # We go to a bit of trouble here to reorder the
510 # indeterminates, so that it's easier to evaluate the
511 # characteristic polynomial at x's coordinates and get back
512 # something in terms of t, which is what we want.
513 S
= PolynomialRing(self
.base_ring(),'t')
517 S
= PolynomialRing(S
, R
.variable_names())
520 return (t
**r
+ sum( a
[k
]*(t
**k
) for k
in range(r
) ))
522 def coordinate_polynomial_ring(self
):
524 The multivariate polynomial ring in which this algebra's
525 :meth:`characteristic_polynomial_of` lives.
529 sage: from mjo.eja.eja_algebra import (HadamardEJA,
530 ....: RealSymmetricEJA)
534 sage: J = HadamardEJA(2)
535 sage: J.coordinate_polynomial_ring()
536 Multivariate Polynomial Ring in X1, X2...
537 sage: J = RealSymmetricEJA(3,field=QQ,orthonormalize=False)
538 sage: J.coordinate_polynomial_ring()
539 Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6...
542 var_names
= tuple( "X%d" % z
for z
in range(1, self
.dimension()+1) )
543 return PolynomialRing(self
.base_ring(), var_names
)
545 def inner_product(self
, x
, y
):
547 The inner product associated with this Euclidean Jordan algebra.
549 Defaults to the trace inner product, but can be overridden by
550 subclasses if they are sure that the necessary properties are
555 sage: from mjo.eja.eja_algebra import (random_eja,
557 ....: BilinearFormEJA)
561 Our inner product is "associative," which means the following for
562 a symmetric bilinear form::
564 sage: set_random_seed()
565 sage: J = random_eja()
566 sage: x,y,z = J.random_elements(3)
567 sage: (x*y).inner_product(z) == y.inner_product(x*z)
572 Ensure that this is the usual inner product for the algebras
575 sage: set_random_seed()
576 sage: J = HadamardEJA.random_instance()
577 sage: x,y = J.random_elements(2)
578 sage: actual = x.inner_product(y)
579 sage: expected = x.to_vector().inner_product(y.to_vector())
580 sage: actual == expected
583 Ensure that this is one-half of the trace inner-product in a
584 BilinearFormEJA that isn't just the reals (when ``n`` isn't
585 one). This is in Faraut and Koranyi, and also my "On the
588 sage: set_random_seed()
589 sage: J = BilinearFormEJA.random_instance()
590 sage: n = J.dimension()
591 sage: x = J.random_element()
592 sage: y = J.random_element()
593 sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2)
596 B
= self
._inner
_product
_matrix
597 return (B
*x
.to_vector()).inner_product(y
.to_vector())
600 def is_trivial(self
):
602 Return whether or not this algebra is trivial.
604 A trivial algebra contains only the zero element.
608 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
613 sage: J = ComplexHermitianEJA(3)
619 sage: J = TrivialEJA()
624 return self
.dimension() == 0
627 def multiplication_table(self
):
629 Return a visual representation of this algebra's multiplication
630 table (on basis elements).
634 sage: from mjo.eja.eja_algebra import JordanSpinEJA
638 sage: J = JordanSpinEJA(4)
639 sage: J.multiplication_table()
640 +----++----+----+----+----+
641 | * || e0 | e1 | e2 | e3 |
642 +====++====+====+====+====+
643 | e0 || e0 | e1 | e2 | e3 |
644 +----++----+----+----+----+
645 | e1 || e1 | e0 | 0 | 0 |
646 +----++----+----+----+----+
647 | e2 || e2 | 0 | e0 | 0 |
648 +----++----+----+----+----+
649 | e3 || e3 | 0 | 0 | e0 |
650 +----++----+----+----+----+
654 # Prepend the header row.
655 M
= [["*"] + list(self
.gens())]
657 # And to each subsequent row, prepend an entry that belongs to
658 # the left-side "header column."
659 M
+= [ [self
.monomial(i
)] + [ self
.product_on_basis(i
,j
)
663 return table(M
, header_row
=True, header_column
=True, frame
=True)
666 def matrix_basis(self
):
668 Return an (often more natural) representation of this algebras
669 basis as an ordered tuple of matrices.
671 Every finite-dimensional Euclidean Jordan Algebra is a, up to
672 Jordan isomorphism, a direct sum of five simple
673 algebras---four of which comprise Hermitian matrices. And the
674 last type of algebra can of course be thought of as `n`-by-`1`
675 column matrices (ambiguusly called column vectors) to avoid
676 special cases. As a result, matrices (and column vectors) are
677 a natural representation format for Euclidean Jordan algebra
680 But, when we construct an algebra from a basis of matrices,
681 those matrix representations are lost in favor of coordinate
682 vectors *with respect to* that basis. We could eventually
683 convert back if we tried hard enough, but having the original
684 representations handy is valuable enough that we simply store
685 them and return them from this method.
687 Why implement this for non-matrix algebras? Avoiding special
688 cases for the :class:`BilinearFormEJA` pays with simplicity in
689 its own right. But mainly, we would like to be able to assume
690 that elements of a :class:`DirectSumEJA` can be displayed
691 nicely, without having to have special classes for direct sums
692 one of whose components was a matrix algebra.
696 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
697 ....: RealSymmetricEJA)
701 sage: J = RealSymmetricEJA(2)
703 Finite family {0: e0, 1: e1, 2: e2}
704 sage: J.matrix_basis()
706 [1 0] [ 0 0.7071067811865475?] [0 0]
707 [0 0], [0.7071067811865475? 0], [0 1]
712 sage: J = JordanSpinEJA(2)
714 Finite family {0: e0, 1: e1}
715 sage: J.matrix_basis()
721 return self
._matrix
_basis
724 def matrix_space(self
):
726 Return the matrix space in which this algebra's elements live, if
727 we think of them as matrices (including column vectors of the
730 Generally this will be an `n`-by-`1` column-vector space,
731 except when the algebra is trivial. There it's `n`-by-`n`
732 (where `n` is zero), to ensure that two elements of the matrix
733 space (empty matrices) can be multiplied.
735 Matrix algebras override this with something more useful.
737 if self
.is_trivial():
738 return MatrixSpace(self
.base_ring(), 0)
740 return self
._matrix
_basis
[0].matrix_space()
746 Return the unit element of this algebra.
750 sage: from mjo.eja.eja_algebra import (HadamardEJA,
755 sage: J = HadamardEJA(5)
757 e0 + e1 + e2 + e3 + e4
761 The identity element acts like the identity::
763 sage: set_random_seed()
764 sage: J = random_eja()
765 sage: x = J.random_element()
766 sage: J.one()*x == x and x*J.one() == x
769 The matrix of the unit element's operator is the identity::
771 sage: set_random_seed()
772 sage: J = random_eja()
773 sage: actual = J.one().operator().matrix()
774 sage: expected = matrix.identity(J.base_ring(), J.dimension())
775 sage: actual == expected
778 Ensure that the cached unit element (often precomputed by
779 hand) agrees with the computed one::
781 sage: set_random_seed()
782 sage: J = random_eja()
783 sage: cached = J.one()
784 sage: J.one.clear_cache()
785 sage: J.one() == cached
789 # We can brute-force compute the matrices of the operators
790 # that correspond to the basis elements of this algebra.
791 # If some linear combination of those basis elements is the
792 # algebra identity, then the same linear combination of
793 # their matrices has to be the identity matrix.
795 # Of course, matrices aren't vectors in sage, so we have to
796 # appeal to the "long vectors" isometry.
797 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
799 # Now we use basic linear algebra to find the coefficients,
800 # of the matrices-as-vectors-linear-combination, which should
801 # work for the original algebra basis too.
802 A
= matrix(self
.base_ring(), oper_vecs
)
804 # We used the isometry on the left-hand side already, but we
805 # still need to do it for the right-hand side. Recall that we
806 # wanted something that summed to the identity matrix.
807 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
809 # Now if there's an identity element in the algebra, this
810 # should work. We solve on the left to avoid having to
811 # transpose the matrix "A".
812 return self
.from_vector(A
.solve_left(b
))
815 def peirce_decomposition(self
, c
):
817 The Peirce decomposition of this algebra relative to the
820 In the future, this can be extended to a complete system of
821 orthogonal idempotents.
825 - ``c`` -- an idempotent of this algebra.
829 A triple (J0, J5, J1) containing two subalgebras and one subspace
832 - ``J0`` -- the algebra on the eigenspace of ``c.operator()``
833 corresponding to the eigenvalue zero.
835 - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()``
836 corresponding to the eigenvalue one-half.
838 - ``J1`` -- the algebra on the eigenspace of ``c.operator()``
839 corresponding to the eigenvalue one.
841 These are the only possible eigenspaces for that operator, and this
842 algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are
843 orthogonal, and are subalgebras of this algebra with the appropriate
848 sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA
852 The canonical example comes from the symmetric matrices, which
853 decompose into diagonal and off-diagonal parts::
855 sage: J = RealSymmetricEJA(3)
856 sage: C = matrix(QQ, [ [1,0,0],
860 sage: J0,J5,J1 = J.peirce_decomposition(c)
862 Euclidean Jordan algebra of dimension 1...
864 Vector space of degree 6 and dimension 2...
866 Euclidean Jordan algebra of dimension 3...
867 sage: J0.one().to_matrix()
871 sage: orig_df = AA.options.display_format
872 sage: AA.options.display_format = 'radical'
873 sage: J.from_vector(J5.basis()[0]).to_matrix()
877 sage: J.from_vector(J5.basis()[1]).to_matrix()
881 sage: AA.options.display_format = orig_df
882 sage: J1.one().to_matrix()
889 Every algebra decomposes trivially with respect to its identity
892 sage: set_random_seed()
893 sage: J = random_eja()
894 sage: J0,J5,J1 = J.peirce_decomposition(J.one())
895 sage: J0.dimension() == 0 and J5.dimension() == 0
897 sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
900 The decomposition is into eigenspaces, and its components are
901 therefore necessarily orthogonal. Moreover, the identity
902 elements in the two subalgebras are the projections onto their
903 respective subspaces of the superalgebra's identity element::
905 sage: set_random_seed()
906 sage: J = random_eja()
907 sage: x = J.random_element()
908 sage: if not J.is_trivial():
909 ....: while x.is_nilpotent():
910 ....: x = J.random_element()
911 sage: c = x.subalgebra_idempotent()
912 sage: J0,J5,J1 = J.peirce_decomposition(c)
914 sage: for (w,y,z) in zip(J0.basis(), J5.basis(), J1.basis()):
915 ....: w = w.superalgebra_element()
916 ....: y = J.from_vector(y)
917 ....: z = z.superalgebra_element()
918 ....: ipsum += w.inner_product(y).abs()
919 ....: ipsum += w.inner_product(z).abs()
920 ....: ipsum += y.inner_product(z).abs()
923 sage: J1(c) == J1.one()
925 sage: J0(J.one() - c) == J0.one()
929 if not c
.is_idempotent():
930 raise ValueError("element is not idempotent: %s" % c
)
932 from mjo
.eja
.eja_subalgebra
import FiniteDimensionalEJASubalgebra
934 # Default these to what they should be if they turn out to be
935 # trivial, because eigenspaces_left() won't return eigenvalues
936 # corresponding to trivial spaces (e.g. it returns only the
937 # eigenspace corresponding to lambda=1 if you take the
938 # decomposition relative to the identity element).
939 trivial
= FiniteDimensionalEJASubalgebra(self
, ())
940 J0
= trivial
# eigenvalue zero
941 J5
= VectorSpace(self
.base_ring(), 0) # eigenvalue one-half
942 J1
= trivial
# eigenvalue one
944 for (eigval
, eigspace
) in c
.operator().matrix().right_eigenspaces():
945 if eigval
== ~
(self
.base_ring()(2)):
948 gens
= tuple( self
.from_vector(b
) for b
in eigspace
.basis() )
949 subalg
= FiniteDimensionalEJASubalgebra(self
,
957 raise ValueError("unexpected eigenvalue: %s" % eigval
)
962 def random_element(self
, thorough
=False):
964 Return a random element of this algebra.
966 Our algebra superclass method only returns a linear
967 combination of at most two basis elements. We instead
968 want the vector space "random element" method that
969 returns a more diverse selection.
973 - ``thorough`` -- (boolean; default False) whether or not we
974 should generate irrational coefficients for the random
975 element when our base ring is irrational; this slows the
976 algebra operations to a crawl, but any truly random method
980 # For a general base ring... maybe we can trust this to do the
981 # right thing? Unlikely, but.
982 V
= self
.vector_space()
983 v
= V
.random_element()
985 if self
.base_ring() is AA
:
986 # The "random element" method of the algebraic reals is
987 # stupid at the moment, and only returns integers between
988 # -2 and 2, inclusive:
990 # https://trac.sagemath.org/ticket/30875
992 # Instead, we implement our own "random vector" method,
993 # and then coerce that into the algebra. We use the vector
994 # space degree here instead of the dimension because a
995 # subalgebra could (for example) be spanned by only two
996 # vectors, each with five coordinates. We need to
997 # generate all five coordinates.
999 v
*= QQbar
.random_element().real()
1001 v
*= QQ
.random_element()
1003 return self
.from_vector(V
.coordinate_vector(v
))
1005 def random_elements(self
, count
, thorough
=False):
1007 Return ``count`` random elements as a tuple.
1011 - ``thorough`` -- (boolean; default False) whether or not we
1012 should generate irrational coefficients for the random
1013 elements when our base ring is irrational; this slows the
1014 algebra operations to a crawl, but any truly random method
1019 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1023 sage: J = JordanSpinEJA(3)
1024 sage: x,y,z = J.random_elements(3)
1025 sage: all( [ x in J, y in J, z in J ])
1027 sage: len( J.random_elements(10) ) == 10
1031 return tuple( self
.random_element(thorough
)
1032 for idx
in range(count
) )
1036 def _charpoly_coefficients(self
):
1038 The `r` polynomial coefficients of the "characteristic polynomial
1041 n
= self
.dimension()
1042 R
= self
.coordinate_polynomial_ring()
1044 F
= R
.fraction_field()
1047 # From a result in my book, these are the entries of the
1048 # basis representation of L_x.
1049 return sum( vars[k
]*self
.monomial(k
).operator().matrix()[i
,j
]
1052 L_x
= matrix(F
, n
, n
, L_x_i_j
)
1055 if self
.rank
.is_in_cache():
1057 # There's no need to pad the system with redundant
1058 # columns if we *know* they'll be redundant.
1061 # Compute an extra power in case the rank is equal to
1062 # the dimension (otherwise, we would stop at x^(r-1)).
1063 x_powers
= [ (L_x
**k
)*self
.one().to_vector()
1064 for k
in range(n
+1) ]
1065 A
= matrix
.column(F
, x_powers
[:n
])
1066 AE
= A
.extended_echelon_form()
1073 # The theory says that only the first "r" coefficients are
1074 # nonzero, and they actually live in the original polynomial
1075 # ring and not the fraction field. We negate them because
1076 # in the actual characteristic polynomial, they get moved
1077 # to the other side where x^r lives.
1078 return -A_rref
.solve_right(E
*b
).change_ring(R
)[:r
]
1083 Return the rank of this EJA.
1085 This is a cached method because we know the rank a priori for
1086 all of the algebras we can construct. Thus we can avoid the
1087 expensive ``_charpoly_coefficients()`` call unless we truly
1088 need to compute the whole characteristic polynomial.
1092 sage: from mjo.eja.eja_algebra import (HadamardEJA,
1093 ....: JordanSpinEJA,
1094 ....: RealSymmetricEJA,
1095 ....: ComplexHermitianEJA,
1096 ....: QuaternionHermitianEJA,
1101 The rank of the Jordan spin algebra is always two::
1103 sage: JordanSpinEJA(2).rank()
1105 sage: JordanSpinEJA(3).rank()
1107 sage: JordanSpinEJA(4).rank()
1110 The rank of the `n`-by-`n` Hermitian real, complex, or
1111 quaternion matrices is `n`::
1113 sage: RealSymmetricEJA(4).rank()
1115 sage: ComplexHermitianEJA(3).rank()
1117 sage: QuaternionHermitianEJA(2).rank()
1122 Ensure that every EJA that we know how to construct has a
1123 positive integer rank, unless the algebra is trivial in
1124 which case its rank will be zero::
1126 sage: set_random_seed()
1127 sage: J = random_eja()
1131 sage: r > 0 or (r == 0 and J.is_trivial())
1134 Ensure that computing the rank actually works, since the ranks
1135 of all simple algebras are known and will be cached by default::
1137 sage: set_random_seed() # long time
1138 sage: J = random_eja() # long time
1139 sage: caches = J.rank() # long time
1140 sage: J.rank.clear_cache() # long time
1141 sage: J.rank() == cached # long time
1145 return len(self
._charpoly
_coefficients
())
1148 def vector_space(self
):
1150 Return the vector space that underlies this algebra.
1154 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1158 sage: J = RealSymmetricEJA(2)
1159 sage: J.vector_space()
1160 Vector space of dimension 3 over...
1163 return self
.zero().to_vector().parent().ambient_vector_space()
1166 Element
= FiniteDimensionalEJAElement
1168 class RationalBasisEJA(FiniteDimensionalEJA
):
1170 New class for algebras whose supplied basis elements have all rational entries.
1174 sage: from mjo.eja.eja_algebra import BilinearFormEJA
1178 The supplied basis is orthonormalized by default::
1180 sage: B = matrix(QQ, [[1, 0, 0], [0, 25, -32], [0, -32, 41]])
1181 sage: J = BilinearFormEJA(B)
1182 sage: J.matrix_basis()
1199 # Abuse the check_field parameter to check that the entries of
1200 # out basis (in ambient coordinates) are in the field QQ.
1201 if not all( all(b_i
in QQ
for b_i
in b
.list()) for b
in basis
):
1202 raise TypeError("basis not rational")
1205 # There's no point in constructing the extra algebra if this
1206 # one is already rational.
1208 # Note: the same Jordan and inner-products work here,
1209 # because they are necessarily defined with respect to
1210 # ambient coordinates and not any particular basis.
1211 self
._rational
_algebra
= FiniteDimensionalEJA(
1216 orthonormalize
=False,
1220 super().__init
__(basis
,
1224 check_field
=check_field
,
1228 def _charpoly_coefficients(self
):
1232 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
1233 ....: JordanSpinEJA)
1237 The base ring of the resulting polynomial coefficients is what
1238 it should be, and not the rationals (unless the algebra was
1239 already over the rationals)::
1241 sage: J = JordanSpinEJA(3)
1242 sage: J._charpoly_coefficients()
1243 (X1^2 - X2^2 - X3^2, -2*X1)
1244 sage: a0 = J._charpoly_coefficients()[0]
1246 Algebraic Real Field
1247 sage: a0.base_ring()
1248 Algebraic Real Field
1251 if self
._rational
_algebra
is None:
1252 # There's no need to construct *another* algebra over the
1253 # rationals if this one is already over the
1254 # rationals. Likewise, if we never orthonormalized our
1255 # basis, we might as well just use the given one.
1256 return super()._charpoly
_coefficients
()
1258 # Do the computation over the rationals. The answer will be
1259 # the same, because all we've done is a change of basis.
1260 # Then, change back from QQ to our real base ring
1261 a
= ( a_i
.change_ring(self
.base_ring())
1262 for a_i
in self
._rational
_algebra
._charpoly
_coefficients
() )
1264 # Now convert the coordinate variables back to the
1265 # deorthonormalized ones.
1266 R
= self
.coordinate_polynomial_ring()
1267 from sage
.modules
.free_module_element
import vector
1268 X
= vector(R
, R
.gens())
1269 BX
= self
._deortho
_matrix
*X
1271 subs_dict
= { X[i]: BX[i] for i in range(len(X)) }
1272 return tuple( a_i
.subs(subs_dict
) for a_i
in a
)
1274 class ConcreteEJA(RationalBasisEJA
):
1276 A class for the Euclidean Jordan algebras that we know by name.
1278 These are the Jordan algebras whose basis, multiplication table,
1279 rank, and so on are known a priori. More to the point, they are
1280 the Euclidean Jordan algebras for which we are able to conjure up
1281 a "random instance."
1285 sage: from mjo.eja.eja_algebra import ConcreteEJA
1289 Our basis is normalized with respect to the algebra's inner
1290 product, unless we specify otherwise::
1292 sage: set_random_seed()
1293 sage: J = ConcreteEJA.random_instance()
1294 sage: all( b.norm() == 1 for b in J.gens() )
1297 Since our basis is orthonormal with respect to the algebra's inner
1298 product, and since we know that this algebra is an EJA, any
1299 left-multiplication operator's matrix will be symmetric because
1300 natural->EJA basis representation is an isometry and within the
1301 EJA the operator is self-adjoint by the Jordan axiom::
1303 sage: set_random_seed()
1304 sage: J = ConcreteEJA.random_instance()
1305 sage: x = J.random_element()
1306 sage: x.operator().is_self_adjoint()
1311 def _max_random_instance_size():
1313 Return an integer "size" that is an upper bound on the size of
1314 this algebra when it is used in a random test
1315 case. Unfortunately, the term "size" is ambiguous -- when
1316 dealing with `R^n` under either the Hadamard or Jordan spin
1317 product, the "size" refers to the dimension `n`. When dealing
1318 with a matrix algebra (real symmetric or complex/quaternion
1319 Hermitian), it refers to the size of the matrix, which is far
1320 less than the dimension of the underlying vector space.
1322 This method must be implemented in each subclass.
1324 raise NotImplementedError
1327 def random_instance(cls
, *args
, **kwargs
):
1329 Return a random instance of this type of algebra.
1331 This method should be implemented in each subclass.
1333 from sage
.misc
.prandom
import choice
1334 eja_class
= choice(cls
.__subclasses
__())
1336 # These all bubble up to the RationalBasisEJA superclass
1337 # constructor, so any (kw)args valid there are also valid
1339 return eja_class
.random_instance(*args
, **kwargs
)
1344 def dimension_over_reals():
1346 The dimension of this matrix's base ring over the reals.
1348 The reals are dimension one over themselves, obviously; that's
1349 just `\mathbb{R}^{1}`. Likewise, the complex numbers `a + bi`
1350 have dimension two. Finally, the quaternions have dimension
1351 four over the reals.
1353 This is used to determine the size of the matrix returned from
1354 :meth:`real_embed`, among other things.
1356 raise NotImplementedError
1359 def real_embed(cls
,M
):
1361 Embed the matrix ``M`` into a space of real matrices.
1363 The matrix ``M`` can have entries in any field at the moment:
1364 the real numbers, complex numbers, or quaternions. And although
1365 they are not a field, we can probably support octonions at some
1366 point, too. This function returns a real matrix that "acts like"
1367 the original with respect to matrix multiplication; i.e.
1369 real_embed(M*N) = real_embed(M)*real_embed(N)
1372 if M
.ncols() != M
.nrows():
1373 raise ValueError("the matrix 'M' must be square")
1378 def real_unembed(cls
,M
):
1380 The inverse of :meth:`real_embed`.
1382 if M
.ncols() != M
.nrows():
1383 raise ValueError("the matrix 'M' must be square")
1384 if not ZZ(M
.nrows()).mod(cls
.dimension_over_reals()).is_zero():
1385 raise ValueError("the matrix 'M' must be a real embedding")
1389 def jordan_product(X
,Y
):
1390 return (X
*Y
+ Y
*X
)/2
1393 def trace_inner_product(cls
,X
,Y
):
1395 Compute the trace inner-product of two real-embeddings.
1399 sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
1400 ....: ComplexHermitianEJA,
1401 ....: QuaternionHermitianEJA)
1405 This gives the same answer as it would if we computed the trace
1406 from the unembedded (original) matrices::
1408 sage: set_random_seed()
1409 sage: J = RealSymmetricEJA.random_instance()
1410 sage: x,y = J.random_elements(2)
1411 sage: Xe = x.to_matrix()
1412 sage: Ye = y.to_matrix()
1413 sage: X = J.real_unembed(Xe)
1414 sage: Y = J.real_unembed(Ye)
1415 sage: expected = (X*Y).trace()
1416 sage: actual = J.trace_inner_product(Xe,Ye)
1417 sage: actual == expected
1422 sage: set_random_seed()
1423 sage: J = ComplexHermitianEJA.random_instance()
1424 sage: x,y = J.random_elements(2)
1425 sage: Xe = x.to_matrix()
1426 sage: Ye = y.to_matrix()
1427 sage: X = J.real_unembed(Xe)
1428 sage: Y = J.real_unembed(Ye)
1429 sage: expected = (X*Y).trace().real()
1430 sage: actual = J.trace_inner_product(Xe,Ye)
1431 sage: actual == expected
1436 sage: set_random_seed()
1437 sage: J = QuaternionHermitianEJA.random_instance()
1438 sage: x,y = J.random_elements(2)
1439 sage: Xe = x.to_matrix()
1440 sage: Ye = y.to_matrix()
1441 sage: X = J.real_unembed(Xe)
1442 sage: Y = J.real_unembed(Ye)
1443 sage: expected = (X*Y).trace().coefficient_tuple()[0]
1444 sage: actual = J.trace_inner_product(Xe,Ye)
1445 sage: actual == expected
1449 Xu
= cls
.real_unembed(X
)
1450 Yu
= cls
.real_unembed(Y
)
1451 tr
= (Xu
*Yu
).trace()
1454 # Works in QQ, AA, RDF, et cetera.
1456 except AttributeError:
1457 # A quaternion doesn't have a real() method, but does
1458 # have coefficient_tuple() method that returns the
1459 # coefficients of 1, i, j, and k -- in that order.
1460 return tr
.coefficient_tuple()[0]
1463 class RealMatrixEJA(MatrixEJA
):
1465 def dimension_over_reals():
1469 class RealSymmetricEJA(ConcreteEJA
, RealMatrixEJA
):
1471 The rank-n simple EJA consisting of real symmetric n-by-n
1472 matrices, the usual symmetric Jordan product, and the trace inner
1473 product. It has dimension `(n^2 + n)/2` over the reals.
1477 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1481 sage: J = RealSymmetricEJA(2)
1482 sage: e0, e1, e2 = J.gens()
1490 In theory, our "field" can be any subfield of the reals::
1492 sage: RealSymmetricEJA(2, field=RDF)
1493 Euclidean Jordan algebra of dimension 3 over Real Double Field
1494 sage: RealSymmetricEJA(2, field=RR)
1495 Euclidean Jordan algebra of dimension 3 over Real Field with
1496 53 bits of precision
1500 The dimension of this algebra is `(n^2 + n) / 2`::
1502 sage: set_random_seed()
1503 sage: n_max = RealSymmetricEJA._max_random_instance_size()
1504 sage: n = ZZ.random_element(1, n_max)
1505 sage: J = RealSymmetricEJA(n)
1506 sage: J.dimension() == (n^2 + n)/2
1509 The Jordan multiplication is what we think it is::
1511 sage: set_random_seed()
1512 sage: J = RealSymmetricEJA.random_instance()
1513 sage: x,y = J.random_elements(2)
1514 sage: actual = (x*y).to_matrix()
1515 sage: X = x.to_matrix()
1516 sage: Y = y.to_matrix()
1517 sage: expected = (X*Y + Y*X)/2
1518 sage: actual == expected
1520 sage: J(expected) == x*y
1523 We can change the generator prefix::
1525 sage: RealSymmetricEJA(3, prefix='q').gens()
1526 (q0, q1, q2, q3, q4, q5)
1528 We can construct the (trivial) algebra of rank zero::
1530 sage: RealSymmetricEJA(0)
1531 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
1535 def _denormalized_basis(cls
, n
):
1537 Return a basis for the space of real symmetric n-by-n matrices.
1541 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1545 sage: set_random_seed()
1546 sage: n = ZZ.random_element(1,5)
1547 sage: B = RealSymmetricEJA._denormalized_basis(n)
1548 sage: all( M.is_symmetric() for M in B)
1552 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1556 for j
in range(i
+1):
1557 Eij
= matrix(ZZ
, n
, lambda k
,l
: k
==i
and l
==j
)
1561 Sij
= Eij
+ Eij
.transpose()
1567 def _max_random_instance_size():
1568 return 4 # Dimension 10
1571 def random_instance(cls
, **kwargs
):
1573 Return a random instance of this type of algebra.
1575 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
1576 return cls(n
, **kwargs
)
1578 def __init__(self
, n
, **kwargs
):
1579 # We know this is a valid EJA, but will double-check
1580 # if the user passes check_axioms=True.
1581 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
1583 super(RealSymmetricEJA
, self
).__init
__(self
._denormalized
_basis
(n
),
1584 self
.jordan_product
,
1585 self
.trace_inner_product
,
1588 # TODO: this could be factored out somehow, but is left here
1589 # because the MatrixEJA is not presently a subclass of the
1590 # FDEJA class that defines rank() and one().
1591 self
.rank
.set_cache(n
)
1592 idV
= matrix
.identity(ZZ
, self
.dimension_over_reals()*n
)
1593 self
.one
.set_cache(self(idV
))
1597 class ComplexMatrixEJA(MatrixEJA
):
1599 def dimension_over_reals():
1603 def real_embed(cls
,M
):
1605 Embed the n-by-n complex matrix ``M`` into the space of real
1606 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1607 bi` to the block matrix ``[[a,b],[-b,a]]``.
1611 sage: from mjo.eja.eja_algebra import ComplexMatrixEJA
1615 sage: F = QuadraticField(-1, 'I')
1616 sage: x1 = F(4 - 2*i)
1617 sage: x2 = F(1 + 2*i)
1620 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1621 sage: ComplexMatrixEJA.real_embed(M)
1630 Embedding is a homomorphism (isomorphism, in fact)::
1632 sage: set_random_seed()
1633 sage: n = ZZ.random_element(3)
1634 sage: F = QuadraticField(-1, 'I')
1635 sage: X = random_matrix(F, n)
1636 sage: Y = random_matrix(F, n)
1637 sage: Xe = ComplexMatrixEJA.real_embed(X)
1638 sage: Ye = ComplexMatrixEJA.real_embed(Y)
1639 sage: XYe = ComplexMatrixEJA.real_embed(X*Y)
1644 super(ComplexMatrixEJA
,cls
).real_embed(M
)
1647 # We don't need any adjoined elements...
1648 field
= M
.base_ring().base_ring()
1652 a
= z
.list()[0] # real part, I guess
1653 b
= z
.list()[1] # imag part, I guess
1654 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1656 return matrix
.block(field
, n
, blocks
)
1660 def real_unembed(cls
,M
):
1662 The inverse of _embed_complex_matrix().
1666 sage: from mjo.eja.eja_algebra import ComplexMatrixEJA
1670 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1671 ....: [-2, 1, -4, 3],
1672 ....: [ 9, 10, 11, 12],
1673 ....: [-10, 9, -12, 11] ])
1674 sage: ComplexMatrixEJA.real_unembed(A)
1676 [ 10*I + 9 12*I + 11]
1680 Unembedding is the inverse of embedding::
1682 sage: set_random_seed()
1683 sage: F = QuadraticField(-1, 'I')
1684 sage: M = random_matrix(F, 3)
1685 sage: Me = ComplexMatrixEJA.real_embed(M)
1686 sage: ComplexMatrixEJA.real_unembed(Me) == M
1690 super(ComplexMatrixEJA
,cls
).real_unembed(M
)
1692 d
= cls
.dimension_over_reals()
1694 # If "M" was normalized, its base ring might have roots
1695 # adjoined and they can stick around after unembedding.
1696 field
= M
.base_ring()
1697 R
= PolynomialRing(field
, 'z')
1700 # Sage doesn't know how to adjoin the complex "i" (the root of
1701 # x^2 + 1) to a field in a general way. Here, we just enumerate
1702 # all of the cases that I have cared to support so far.
1704 # Sage doesn't know how to embed AA into QQbar, i.e. how
1705 # to adjoin sqrt(-1) to AA.
1707 elif not field
.is_exact():
1709 F
= field
.complex_field()
1711 # Works for QQ and... maybe some other fields.
1712 F
= field
.extension(z
**2 + 1, 'I', embedding
=CLF(-1).sqrt())
1715 # Go top-left to bottom-right (reading order), converting every
1716 # 2-by-2 block we see to a single complex element.
1718 for k
in range(n
/d
):
1719 for j
in range(n
/d
):
1720 submat
= M
[d
*k
:d
*k
+d
,d
*j
:d
*j
+d
]
1721 if submat
[0,0] != submat
[1,1]:
1722 raise ValueError('bad on-diagonal submatrix')
1723 if submat
[0,1] != -submat
[1,0]:
1724 raise ValueError('bad off-diagonal submatrix')
1725 z
= submat
[0,0] + submat
[0,1]*i
1728 return matrix(F
, n
/d
, elements
)
1731 class ComplexHermitianEJA(ConcreteEJA
, ComplexMatrixEJA
):
1733 The rank-n simple EJA consisting of complex Hermitian n-by-n
1734 matrices over the real numbers, the usual symmetric Jordan product,
1735 and the real-part-of-trace inner product. It has dimension `n^2` over
1740 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1744 In theory, our "field" can be any subfield of the reals::
1746 sage: ComplexHermitianEJA(2, field=RDF)
1747 Euclidean Jordan algebra of dimension 4 over Real Double Field
1748 sage: ComplexHermitianEJA(2, field=RR)
1749 Euclidean Jordan algebra of dimension 4 over Real Field with
1750 53 bits of precision
1754 The dimension of this algebra is `n^2`::
1756 sage: set_random_seed()
1757 sage: n_max = ComplexHermitianEJA._max_random_instance_size()
1758 sage: n = ZZ.random_element(1, n_max)
1759 sage: J = ComplexHermitianEJA(n)
1760 sage: J.dimension() == n^2
1763 The Jordan multiplication is what we think it is::
1765 sage: set_random_seed()
1766 sage: J = ComplexHermitianEJA.random_instance()
1767 sage: x,y = J.random_elements(2)
1768 sage: actual = (x*y).to_matrix()
1769 sage: X = x.to_matrix()
1770 sage: Y = y.to_matrix()
1771 sage: expected = (X*Y + Y*X)/2
1772 sage: actual == expected
1774 sage: J(expected) == x*y
1777 We can change the generator prefix::
1779 sage: ComplexHermitianEJA(2, prefix='z').gens()
1782 We can construct the (trivial) algebra of rank zero::
1784 sage: ComplexHermitianEJA(0)
1785 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
1790 def _denormalized_basis(cls
, n
):
1792 Returns a basis for the space of complex Hermitian n-by-n matrices.
1794 Why do we embed these? Basically, because all of numerical linear
1795 algebra assumes that you're working with vectors consisting of `n`
1796 entries from a field and scalars from the same field. There's no way
1797 to tell SageMath that (for example) the vectors contain complex
1798 numbers, while the scalar field is real.
1802 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1806 sage: set_random_seed()
1807 sage: n = ZZ.random_element(1,5)
1808 sage: field = QuadraticField(2, 'sqrt2')
1809 sage: B = ComplexHermitianEJA._denormalized_basis(n)
1810 sage: all( M.is_symmetric() for M in B)
1815 R
= PolynomialRing(field
, 'z')
1817 F
= field
.extension(z
**2 + 1, 'I')
1820 # This is like the symmetric case, but we need to be careful:
1822 # * We want conjugate-symmetry, not just symmetry.
1823 # * The diagonal will (as a result) be real.
1827 for j
in range(i
+1):
1828 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1830 Sij
= cls
.real_embed(Eij
)
1833 # The second one has a minus because it's conjugated.
1834 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1836 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1839 # Since we embedded these, we can drop back to the "field" that we
1840 # started with instead of the complex extension "F".
1841 return tuple( s
.change_ring(field
) for s
in S
)
1844 def __init__(self
, n
, **kwargs
):
1845 # We know this is a valid EJA, but will double-check
1846 # if the user passes check_axioms=True.
1847 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
1849 super(ComplexHermitianEJA
, self
).__init
__(self
._denormalized
_basis
(n
),
1850 self
.jordan_product
,
1851 self
.trace_inner_product
,
1853 # TODO: this could be factored out somehow, but is left here
1854 # because the MatrixEJA is not presently a subclass of the
1855 # FDEJA class that defines rank() and one().
1856 self
.rank
.set_cache(n
)
1857 idV
= matrix
.identity(ZZ
, self
.dimension_over_reals()*n
)
1858 self
.one
.set_cache(self(idV
))
1861 def _max_random_instance_size():
1862 return 3 # Dimension 9
1865 def random_instance(cls
, **kwargs
):
1867 Return a random instance of this type of algebra.
1869 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
1870 return cls(n
, **kwargs
)
1872 class QuaternionMatrixEJA(MatrixEJA
):
1874 def dimension_over_reals():
1878 def real_embed(cls
,M
):
1880 Embed the n-by-n quaternion matrix ``M`` into the space of real
1881 matrices of size 4n-by-4n by first sending each quaternion entry `z
1882 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1883 c+di],[-c + di, a-bi]]`, and then embedding those into a real
1888 sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA
1892 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1893 sage: i,j,k = Q.gens()
1894 sage: x = 1 + 2*i + 3*j + 4*k
1895 sage: M = matrix(Q, 1, [[x]])
1896 sage: QuaternionMatrixEJA.real_embed(M)
1902 Embedding is a homomorphism (isomorphism, in fact)::
1904 sage: set_random_seed()
1905 sage: n = ZZ.random_element(2)
1906 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1907 sage: X = random_matrix(Q, n)
1908 sage: Y = random_matrix(Q, n)
1909 sage: Xe = QuaternionMatrixEJA.real_embed(X)
1910 sage: Ye = QuaternionMatrixEJA.real_embed(Y)
1911 sage: XYe = QuaternionMatrixEJA.real_embed(X*Y)
1916 super(QuaternionMatrixEJA
,cls
).real_embed(M
)
1917 quaternions
= M
.base_ring()
1920 F
= QuadraticField(-1, 'I')
1925 t
= z
.coefficient_tuple()
1930 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1931 [-c
+ d
*i
, a
- b
*i
]])
1932 realM
= ComplexMatrixEJA
.real_embed(cplxM
)
1933 blocks
.append(realM
)
1935 # We should have real entries by now, so use the realest field
1936 # we've got for the return value.
1937 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1942 def real_unembed(cls
,M
):
1944 The inverse of _embed_quaternion_matrix().
1948 sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA
1952 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1953 ....: [-2, 1, -4, 3],
1954 ....: [-3, 4, 1, -2],
1955 ....: [-4, -3, 2, 1]])
1956 sage: QuaternionMatrixEJA.real_unembed(M)
1957 [1 + 2*i + 3*j + 4*k]
1961 Unembedding is the inverse of embedding::
1963 sage: set_random_seed()
1964 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1965 sage: M = random_matrix(Q, 3)
1966 sage: Me = QuaternionMatrixEJA.real_embed(M)
1967 sage: QuaternionMatrixEJA.real_unembed(Me) == M
1971 super(QuaternionMatrixEJA
,cls
).real_unembed(M
)
1973 d
= cls
.dimension_over_reals()
1975 # Use the base ring of the matrix to ensure that its entries can be
1976 # multiplied by elements of the quaternion algebra.
1977 field
= M
.base_ring()
1978 Q
= QuaternionAlgebra(field
,-1,-1)
1981 # Go top-left to bottom-right (reading order), converting every
1982 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1985 for l
in range(n
/d
):
1986 for m
in range(n
/d
):
1987 submat
= ComplexMatrixEJA
.real_unembed(
1988 M
[d
*l
:d
*l
+d
,d
*m
:d
*m
+d
] )
1989 if submat
[0,0] != submat
[1,1].conjugate():
1990 raise ValueError('bad on-diagonal submatrix')
1991 if submat
[0,1] != -submat
[1,0].conjugate():
1992 raise ValueError('bad off-diagonal submatrix')
1993 z
= submat
[0,0].real()
1994 z
+= submat
[0,0].imag()*i
1995 z
+= submat
[0,1].real()*j
1996 z
+= submat
[0,1].imag()*k
1999 return matrix(Q
, n
/d
, elements
)
2002 class QuaternionHermitianEJA(ConcreteEJA
, QuaternionMatrixEJA
):
2004 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
2005 matrices, the usual symmetric Jordan product, and the
2006 real-part-of-trace inner product. It has dimension `2n^2 - n` over
2011 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
2015 In theory, our "field" can be any subfield of the reals::
2017 sage: QuaternionHermitianEJA(2, field=RDF)
2018 Euclidean Jordan algebra of dimension 6 over Real Double Field
2019 sage: QuaternionHermitianEJA(2, field=RR)
2020 Euclidean Jordan algebra of dimension 6 over Real Field with
2021 53 bits of precision
2025 The dimension of this algebra is `2*n^2 - n`::
2027 sage: set_random_seed()
2028 sage: n_max = QuaternionHermitianEJA._max_random_instance_size()
2029 sage: n = ZZ.random_element(1, n_max)
2030 sage: J = QuaternionHermitianEJA(n)
2031 sage: J.dimension() == 2*(n^2) - n
2034 The Jordan multiplication is what we think it is::
2036 sage: set_random_seed()
2037 sage: J = QuaternionHermitianEJA.random_instance()
2038 sage: x,y = J.random_elements(2)
2039 sage: actual = (x*y).to_matrix()
2040 sage: X = x.to_matrix()
2041 sage: Y = y.to_matrix()
2042 sage: expected = (X*Y + Y*X)/2
2043 sage: actual == expected
2045 sage: J(expected) == x*y
2048 We can change the generator prefix::
2050 sage: QuaternionHermitianEJA(2, prefix='a').gens()
2051 (a0, a1, a2, a3, a4, a5)
2053 We can construct the (trivial) algebra of rank zero::
2055 sage: QuaternionHermitianEJA(0)
2056 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
2060 def _denormalized_basis(cls
, n
):
2062 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
2064 Why do we embed these? Basically, because all of numerical
2065 linear algebra assumes that you're working with vectors consisting
2066 of `n` entries from a field and scalars from the same field. There's
2067 no way to tell SageMath that (for example) the vectors contain
2068 complex numbers, while the scalar field is real.
2072 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
2076 sage: set_random_seed()
2077 sage: n = ZZ.random_element(1,5)
2078 sage: B = QuaternionHermitianEJA._denormalized_basis(n)
2079 sage: all( M.is_symmetric() for M in B )
2084 Q
= QuaternionAlgebra(QQ
,-1,-1)
2087 # This is like the symmetric case, but we need to be careful:
2089 # * We want conjugate-symmetry, not just symmetry.
2090 # * The diagonal will (as a result) be real.
2094 for j
in range(i
+1):
2095 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
2097 Sij
= cls
.real_embed(Eij
)
2100 # The second, third, and fourth ones have a minus
2101 # because they're conjugated.
2102 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
2104 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
2106 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
2108 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
2111 # Since we embedded these, we can drop back to the "field" that we
2112 # started with instead of the quaternion algebra "Q".
2113 return tuple( s
.change_ring(field
) for s
in S
)
2116 def __init__(self
, n
, **kwargs
):
2117 # We know this is a valid EJA, but will double-check
2118 # if the user passes check_axioms=True.
2119 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2121 super(QuaternionHermitianEJA
, self
).__init
__(self
._denormalized
_basis
(n
),
2122 self
.jordan_product
,
2123 self
.trace_inner_product
,
2125 # TODO: this could be factored out somehow, but is left here
2126 # because the MatrixEJA is not presently a subclass of the
2127 # FDEJA class that defines rank() and one().
2128 self
.rank
.set_cache(n
)
2129 idV
= matrix
.identity(ZZ
, self
.dimension_over_reals()*n
)
2130 self
.one
.set_cache(self(idV
))
2134 def _max_random_instance_size():
2136 The maximum rank of a random QuaternionHermitianEJA.
2138 return 2 # Dimension 6
2141 def random_instance(cls
, **kwargs
):
2143 Return a random instance of this type of algebra.
2145 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2146 return cls(n
, **kwargs
)
2149 class HadamardEJA(ConcreteEJA
):
2151 Return the Euclidean Jordan Algebra corresponding to the set
2152 `R^n` under the Hadamard product.
2154 Note: this is nothing more than the Cartesian product of ``n``
2155 copies of the spin algebra. Once Cartesian product algebras
2156 are implemented, this can go.
2160 sage: from mjo.eja.eja_algebra import HadamardEJA
2164 This multiplication table can be verified by hand::
2166 sage: J = HadamardEJA(3)
2167 sage: e0,e1,e2 = J.gens()
2183 We can change the generator prefix::
2185 sage: HadamardEJA(3, prefix='r').gens()
2189 def __init__(self
, n
, **kwargs
):
2191 jordan_product
= lambda x
,y
: x
2192 inner_product
= lambda x
,y
: x
2194 def jordan_product(x
,y
):
2196 return P( xi
*yi
for (xi
,yi
) in zip(x
,y
) )
2198 def inner_product(x
,y
):
2201 # New defaults for keyword arguments. Don't orthonormalize
2202 # because our basis is already orthonormal with respect to our
2203 # inner-product. Don't check the axioms, because we know this
2204 # is a valid EJA... but do double-check if the user passes
2205 # check_axioms=True. Note: we DON'T override the "check_field"
2206 # default here, because the user can pass in a field!
2207 if "orthonormalize" not in kwargs
: kwargs
["orthonormalize"] = False
2208 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2210 column_basis
= tuple( b
.column() for b
in FreeModule(ZZ
, n
).basis() )
2211 super().__init
__(column_basis
, jordan_product
, inner_product
, **kwargs
)
2212 self
.rank
.set_cache(n
)
2215 self
.one
.set_cache( self
.zero() )
2217 self
.one
.set_cache( sum(self
.gens()) )
2220 def _max_random_instance_size():
2222 The maximum dimension of a random HadamardEJA.
2227 def random_instance(cls
, **kwargs
):
2229 Return a random instance of this type of algebra.
2231 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2232 return cls(n
, **kwargs
)
2235 class BilinearFormEJA(ConcreteEJA
):
2237 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2238 with the half-trace inner product and jordan product ``x*y =
2239 (<Bx,y>,y_bar>, x0*y_bar + y0*x_bar)`` where `B = 1 \times B22` is
2240 a symmetric positive-definite "bilinear form" matrix. Its
2241 dimension is the size of `B`, and it has rank two in dimensions
2242 larger than two. It reduces to the ``JordanSpinEJA`` when `B` is
2243 the identity matrix of order ``n``.
2245 We insist that the one-by-one upper-left identity block of `B` be
2246 passed in as well so that we can be passed a matrix of size zero
2247 to construct a trivial algebra.
2251 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
2252 ....: JordanSpinEJA)
2256 When no bilinear form is specified, the identity matrix is used,
2257 and the resulting algebra is the Jordan spin algebra::
2259 sage: B = matrix.identity(AA,3)
2260 sage: J0 = BilinearFormEJA(B)
2261 sage: J1 = JordanSpinEJA(3)
2262 sage: J0.multiplication_table() == J0.multiplication_table()
2265 An error is raised if the matrix `B` does not correspond to a
2266 positive-definite bilinear form::
2268 sage: B = matrix.random(QQ,2,3)
2269 sage: J = BilinearFormEJA(B)
2270 Traceback (most recent call last):
2272 ValueError: bilinear form is not positive-definite
2273 sage: B = matrix.zero(QQ,3)
2274 sage: J = BilinearFormEJA(B)
2275 Traceback (most recent call last):
2277 ValueError: bilinear form is not positive-definite
2281 We can create a zero-dimensional algebra::
2283 sage: B = matrix.identity(AA,0)
2284 sage: J = BilinearFormEJA(B)
2288 We can check the multiplication condition given in the Jordan, von
2289 Neumann, and Wigner paper (and also discussed on my "On the
2290 symmetry..." paper). Note that this relies heavily on the standard
2291 choice of basis, as does anything utilizing the bilinear form
2292 matrix. We opt not to orthonormalize the basis, because if we
2293 did, we would have to normalize the `s_{i}` in a similar manner::
2295 sage: set_random_seed()
2296 sage: n = ZZ.random_element(5)
2297 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
2298 sage: B11 = matrix.identity(QQ,1)
2299 sage: B22 = M.transpose()*M
2300 sage: B = block_matrix(2,2,[ [B11,0 ],
2302 sage: J = BilinearFormEJA(B, orthonormalize=False)
2303 sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
2304 sage: V = J.vector_space()
2305 sage: sis = [ J( V([0] + (M.inverse()*ei).list()).column() )
2306 ....: for ei in eis ]
2307 sage: actual = [ sis[i]*sis[j]
2308 ....: for i in range(n-1)
2309 ....: for j in range(n-1) ]
2310 sage: expected = [ J.one() if i == j else J.zero()
2311 ....: for i in range(n-1)
2312 ....: for j in range(n-1) ]
2313 sage: actual == expected
2317 def __init__(self
, B
, **kwargs
):
2318 # The matrix "B" is supplied by the user in most cases,
2319 # so it makes sense to check whether or not its positive-
2320 # definite unless we are specifically asked not to...
2321 if ("check_axioms" not in kwargs
) or kwargs
["check_axioms"]:
2322 if not B
.is_positive_definite():
2323 raise ValueError("bilinear form is not positive-definite")
2325 # However, all of the other data for this EJA is computed
2326 # by us in manner that guarantees the axioms are
2327 # satisfied. So, again, unless we are specifically asked to
2328 # verify things, we'll skip the rest of the checks.
2329 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2331 def inner_product(x
,y
):
2332 return (y
.T
*B
*x
)[0,0]
2334 def jordan_product(x
,y
):
2340 z0
= inner_product(y
,x
)
2341 zbar
= y0
*xbar
+ x0
*ybar
2342 return P([z0
] + zbar
.list())
2345 column_basis
= tuple( b
.column() for b
in FreeModule(ZZ
, n
).basis() )
2346 super(BilinearFormEJA
, self
).__init
__(column_basis
,
2351 # The rank of this algebra is two, unless we're in a
2352 # one-dimensional ambient space (because the rank is bounded
2353 # by the ambient dimension).
2354 self
.rank
.set_cache(min(n
,2))
2357 self
.one
.set_cache( self
.zero() )
2359 self
.one
.set_cache( self
.monomial(0) )
2362 def _max_random_instance_size():
2364 The maximum dimension of a random BilinearFormEJA.
2369 def random_instance(cls
, **kwargs
):
2371 Return a random instance of this algebra.
2373 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2375 B
= matrix
.identity(ZZ
, n
)
2376 return cls(B
, **kwargs
)
2378 B11
= matrix
.identity(ZZ
, 1)
2379 M
= matrix
.random(ZZ
, n
-1)
2380 I
= matrix
.identity(ZZ
, n
-1)
2382 while alpha
.is_zero():
2383 alpha
= ZZ
.random_element().abs()
2384 B22
= M
.transpose()*M
+ alpha
*I
2386 from sage
.matrix
.special
import block_matrix
2387 B
= block_matrix(2,2, [ [B11
, ZZ(0) ],
2390 return cls(B
, **kwargs
)
2393 class JordanSpinEJA(BilinearFormEJA
):
2395 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2396 with the usual inner product and jordan product ``x*y =
2397 (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2402 sage: from mjo.eja.eja_algebra import JordanSpinEJA
2406 This multiplication table can be verified by hand::
2408 sage: J = JordanSpinEJA(4)
2409 sage: e0,e1,e2,e3 = J.gens()
2425 We can change the generator prefix::
2427 sage: JordanSpinEJA(2, prefix='B').gens()
2432 Ensure that we have the usual inner product on `R^n`::
2434 sage: set_random_seed()
2435 sage: J = JordanSpinEJA.random_instance()
2436 sage: x,y = J.random_elements(2)
2437 sage: actual = x.inner_product(y)
2438 sage: expected = x.to_vector().inner_product(y.to_vector())
2439 sage: actual == expected
2443 def __init__(self
, n
, **kwargs
):
2444 # This is a special case of the BilinearFormEJA with the
2445 # identity matrix as its bilinear form.
2446 B
= matrix
.identity(ZZ
, n
)
2448 # Don't orthonormalize because our basis is already
2449 # orthonormal with respect to our inner-product.
2450 if "orthonormalize" not in kwargs
: kwargs
["orthonormalize"] = False
2452 # But also don't pass check_field=False here, because the user
2453 # can pass in a field!
2454 super(JordanSpinEJA
, self
).__init
__(B
, **kwargs
)
2457 def _max_random_instance_size():
2459 The maximum dimension of a random JordanSpinEJA.
2464 def random_instance(cls
, **kwargs
):
2466 Return a random instance of this type of algebra.
2468 Needed here to override the implementation for ``BilinearFormEJA``.
2470 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2471 return cls(n
, **kwargs
)
2474 class TrivialEJA(ConcreteEJA
):
2476 The trivial Euclidean Jordan algebra consisting of only a zero element.
2480 sage: from mjo.eja.eja_algebra import TrivialEJA
2484 sage: J = TrivialEJA()
2491 sage: 7*J.one()*12*J.one()
2493 sage: J.one().inner_product(J.one())
2495 sage: J.one().norm()
2497 sage: J.one().subalgebra_generated_by()
2498 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
2503 def __init__(self
, **kwargs
):
2504 jordan_product
= lambda x
,y
: x
2505 inner_product
= lambda x
,y
: 0
2508 # New defaults for keyword arguments
2509 if "orthonormalize" not in kwargs
: kwargs
["orthonormalize"] = False
2510 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2512 super(TrivialEJA
, self
).__init
__(basis
,
2516 # The rank is zero using my definition, namely the dimension of the
2517 # largest subalgebra generated by any element.
2518 self
.rank
.set_cache(0)
2519 self
.one
.set_cache( self
.zero() )
2522 def random_instance(cls
, **kwargs
):
2523 # We don't take a "size" argument so the superclass method is
2524 # inappropriate for us.
2525 return cls(**kwargs
)
2527 # class DirectSumEJA(ConcreteEJA):
2529 # The external (orthogonal) direct sum of two other Euclidean Jordan
2530 # algebras. Essentially the Cartesian product of its two factors.
2531 # Every Euclidean Jordan algebra decomposes into an orthogonal
2532 # direct sum of simple Euclidean Jordan algebras, so no generality
2533 # is lost by providing only this construction.
2537 # sage: from mjo.eja.eja_algebra import (random_eja,
2538 # ....: HadamardEJA,
2539 # ....: RealSymmetricEJA,
2540 # ....: DirectSumEJA)
2544 # sage: J1 = HadamardEJA(2)
2545 # sage: J2 = RealSymmetricEJA(3)
2546 # sage: J = DirectSumEJA(J1,J2)
2547 # sage: J.dimension()
2554 # The external direct sum construction is only valid when the two factors
2555 # have the same base ring; an error is raised otherwise::
2557 # sage: set_random_seed()
2558 # sage: J1 = random_eja(field=AA)
2559 # sage: J2 = random_eja(field=QQ,orthonormalize=False)
2560 # sage: J = DirectSumEJA(J1,J2)
2561 # Traceback (most recent call last):
2563 # ValueError: algebras must share the same base field
2566 # def __init__(self, J1, J2, **kwargs):
2567 # if J1.base_ring() != J2.base_ring():
2568 # raise ValueError("algebras must share the same base field")
2569 # field = J1.base_ring()
2571 # self._factors = (J1, J2)
2572 # n1 = J1.dimension()
2573 # n2 = J2.dimension()
2575 # V = VectorSpace(field, n)
2576 # mult_table = [ [ V.zero() for j in range(i+1) ]
2577 # for i in range(n) ]
2578 # for i in range(n1):
2579 # for j in range(i+1):
2580 # p = (J1.monomial(i)*J1.monomial(j)).to_vector()
2581 # mult_table[i][j] = V(p.list() + [field.zero()]*n2)
2583 # for i in range(n2):
2584 # for j in range(i+1):
2585 # p = (J2.monomial(i)*J2.monomial(j)).to_vector()
2586 # mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list())
2588 # # TODO: build the IP table here from the two constituent IP
2589 # # matrices (it'll be block diagonal, I think).
2590 # ip_table = [ [ field.zero() for j in range(i+1) ]
2591 # for i in range(n) ]
2592 # super(DirectSumEJA, self).__init__(field,
2595 # check_axioms=False,
2597 # self.rank.set_cache(J1.rank() + J2.rank())
2600 # def factors(self):
2602 # Return the pair of this algebra's factors.
2606 # sage: from mjo.eja.eja_algebra import (HadamardEJA,
2607 # ....: JordanSpinEJA,
2608 # ....: DirectSumEJA)
2612 # sage: J1 = HadamardEJA(2, field=QQ)
2613 # sage: J2 = JordanSpinEJA(3, field=QQ)
2614 # sage: J = DirectSumEJA(J1,J2)
2616 # (Euclidean Jordan algebra of dimension 2 over Rational Field,
2617 # Euclidean Jordan algebra of dimension 3 over Rational Field)
2620 # return self._factors
2622 # def projections(self):
2624 # Return a pair of projections onto this algebra's factors.
2628 # sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
2629 # ....: ComplexHermitianEJA,
2630 # ....: DirectSumEJA)
2634 # sage: J1 = JordanSpinEJA(2)
2635 # sage: J2 = ComplexHermitianEJA(2)
2636 # sage: J = DirectSumEJA(J1,J2)
2637 # sage: (pi_left, pi_right) = J.projections()
2638 # sage: J.one().to_vector()
2639 # (1, 0, 1, 0, 0, 1)
2640 # sage: pi_left(J.one()).to_vector()
2642 # sage: pi_right(J.one()).to_vector()
2646 # (J1,J2) = self.factors()
2647 # m = J1.dimension()
2648 # n = J2.dimension()
2649 # V_basis = self.vector_space().basis()
2650 # # Need to specify the dimensions explicitly so that we don't
2651 # # wind up with a zero-by-zero matrix when we want e.g. a
2652 # # zero-by-two matrix (important for composing things).
2653 # P1 = matrix(self.base_ring(), m, m+n, V_basis[:m])
2654 # P2 = matrix(self.base_ring(), n, m+n, V_basis[m:])
2655 # pi_left = FiniteDimensionalEJAOperator(self,J1,P1)
2656 # pi_right = FiniteDimensionalEJAOperator(self,J2,P2)
2657 # return (pi_left, pi_right)
2659 # def inclusions(self):
2661 # Return the pair of inclusion maps from our factors into us.
2665 # sage: from mjo.eja.eja_algebra import (random_eja,
2666 # ....: JordanSpinEJA,
2667 # ....: RealSymmetricEJA,
2668 # ....: DirectSumEJA)
2672 # sage: J1 = JordanSpinEJA(3)
2673 # sage: J2 = RealSymmetricEJA(2)
2674 # sage: J = DirectSumEJA(J1,J2)
2675 # sage: (iota_left, iota_right) = J.inclusions()
2676 # sage: iota_left(J1.zero()) == J.zero()
2678 # sage: iota_right(J2.zero()) == J.zero()
2680 # sage: J1.one().to_vector()
2682 # sage: iota_left(J1.one()).to_vector()
2683 # (1, 0, 0, 0, 0, 0)
2684 # sage: J2.one().to_vector()
2686 # sage: iota_right(J2.one()).to_vector()
2687 # (0, 0, 0, 1, 0, 1)
2688 # sage: J.one().to_vector()
2689 # (1, 0, 0, 1, 0, 1)
2693 # Composing a projection with the corresponding inclusion should
2694 # produce the identity map, and mismatching them should produce
2697 # sage: set_random_seed()
2698 # sage: J1 = random_eja()
2699 # sage: J2 = random_eja()
2700 # sage: J = DirectSumEJA(J1,J2)
2701 # sage: (iota_left, iota_right) = J.inclusions()
2702 # sage: (pi_left, pi_right) = J.projections()
2703 # sage: pi_left*iota_left == J1.one().operator()
2705 # sage: pi_right*iota_right == J2.one().operator()
2707 # sage: (pi_left*iota_right).is_zero()
2709 # sage: (pi_right*iota_left).is_zero()
2713 # (J1,J2) = self.factors()
2714 # m = J1.dimension()
2715 # n = J2.dimension()
2716 # V_basis = self.vector_space().basis()
2717 # # Need to specify the dimensions explicitly so that we don't
2718 # # wind up with a zero-by-zero matrix when we want e.g. a
2719 # # two-by-zero matrix (important for composing things).
2720 # I1 = matrix.column(self.base_ring(), m, m+n, V_basis[:m])
2721 # I2 = matrix.column(self.base_ring(), n, m+n, V_basis[m:])
2722 # iota_left = FiniteDimensionalEJAOperator(J1,self,I1)
2723 # iota_right = FiniteDimensionalEJAOperator(J2,self,I2)
2724 # return (iota_left, iota_right)
2726 # def inner_product(self, x, y):
2728 # The standard Cartesian inner-product.
2730 # We project ``x`` and ``y`` onto our factors, and add up the
2731 # inner-products from the subalgebras.
2736 # sage: from mjo.eja.eja_algebra import (HadamardEJA,
2737 # ....: QuaternionHermitianEJA,
2738 # ....: DirectSumEJA)
2742 # sage: J1 = HadamardEJA(3,field=QQ)
2743 # sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
2744 # sage: J = DirectSumEJA(J1,J2)
2745 # sage: x1 = J1.one()
2747 # sage: y1 = J2.one()
2749 # sage: x1.inner_product(x2)
2751 # sage: y1.inner_product(y2)
2753 # sage: J.one().inner_product(J.one())
2757 # (pi_left, pi_right) = self.projections()
2763 # return (x1.inner_product(y1) + x2.inner_product(y2))
2767 random_eja
= ConcreteEJA
.random_instance