2 Representations and constructions for Euclidean Jordan algebras.
4 A Euclidean Jordan algebra is a Jordan algebra that has some
7 1. It is finite-dimensional.
8 2. Its scalar field is the real numbers.
9 3a. An inner product is defined on it, and...
10 3b. That inner product is compatible with the Jordan product
11 in the sense that `<x*y,z> = <y,x*z>` for all elements
12 `x,y,z` in the algebra.
14 Every Euclidean Jordan algebra is formally-real: for any two elements
15 `x` and `y` in the algebra, `x^{2} + y^{2} = 0` implies that `x = y =
16 0`. Conversely, every finite-dimensional formally-real Jordan algebra
17 can be made into a Euclidean Jordan algebra with an appropriate choice
20 Formally-real Jordan algebras were originally studied as a framework
21 for quantum mechanics. Today, Euclidean Jordan algebras are crucial in
22 symmetric cone optimization, since every symmetric cone arises as the
23 cone of squares in some Euclidean Jordan algebra.
25 It is known that every Euclidean Jordan algebra decomposes into an
26 orthogonal direct sum (essentially, a Cartesian product) of simple
27 algebras, and that moreover, up to Jordan-algebra isomorphism, there
28 are only five families of simple algebras. We provide constructions
29 for these simple algebras:
31 * :class:`BilinearFormEJA`
32 * :class:`RealSymmetricEJA`
33 * :class:`ComplexHermitianEJA`
34 * :class:`QuaternionHermitianEJA`
35 * :class:`OctonionHermitianEJA`
37 In addition to these, we provide two other example constructions,
39 * :class:`JordanSpinEJA`
40 * :class:`HadamardEJA`
44 The Jordan spin algebra is a bilinear form algebra where the bilinear
45 form is the identity. The Hadamard EJA is simply a Cartesian product
46 of one-dimensional spin algebras. The Albert EJA is simply a special
47 case of the :class:`OctonionHermitianEJA` where the matrices are
48 three-by-three and the resulting space has dimension 27. And
49 last/least, the trivial EJA is exactly what you think it is; it could
50 also be obtained by constructing a dimension-zero instance of any of
51 the other algebras. Cartesian products of these are also supported
52 using the usual ``cartesian_product()`` function; as a result, we
53 support (up to isomorphism) all Euclidean Jordan algebras.
57 sage: from mjo.eja.eja_algebra import random_eja
62 Euclidean Jordan algebra of dimension...
65 from sage
.algebras
.quatalg
.quaternion_algebra
import QuaternionAlgebra
66 from sage
.categories
.magmatic_algebras
import MagmaticAlgebras
67 from sage
.categories
.sets_cat
import cartesian_product
68 from sage
.combinat
.free_module
import CombinatorialFreeModule
69 from sage
.matrix
.constructor
import matrix
70 from sage
.matrix
.matrix_space
import MatrixSpace
71 from sage
.misc
.cachefunc
import cached_method
72 from sage
.misc
.table
import table
73 from sage
.modules
.free_module
import FreeModule
, VectorSpace
74 from sage
.rings
.all
import (ZZ
, QQ
, AA
, QQbar
, RR
, RLF
, CLF
,
77 from mjo
.eja
.eja_element
import FiniteDimensionalEJAElement
78 from mjo
.eja
.eja_operator
import FiniteDimensionalEJAOperator
79 from mjo
.eja
.eja_utils
import _all2list
, _mat2vec
81 class FiniteDimensionalEJA(CombinatorialFreeModule
):
83 A finite-dimensional Euclidean Jordan algebra.
87 - ``basis`` -- a tuple; a tuple of basis elements in "matrix
88 form," which must be the same form as the arguments to
89 ``jordan_product`` and ``inner_product``. In reality, "matrix
90 form" can be either vectors, matrices, or a Cartesian product
91 (ordered tuple) of vectors or matrices. All of these would
92 ideally be vector spaces in sage with no special-casing
93 needed; but in reality we turn vectors into column-matrices
94 and Cartesian products `(a,b)` into column matrices
95 `(a,b)^{T}` after converting `a` and `b` themselves.
97 - ``jordan_product`` -- a function; afunction of two ``basis``
98 elements (in matrix form) that returns their jordan product,
99 also in matrix form; this will be applied to ``basis`` to
100 compute a multiplication table for the algebra.
102 - ``inner_product`` -- a function; a function of two ``basis``
103 elements (in matrix form) that returns their inner
104 product. This will be applied to ``basis`` to compute an
105 inner-product table (basically a matrix) for this algebra.
107 - ``matrix_space`` -- the space that your matrix basis lives in,
108 or ``None`` (the default). So long as your basis does not have
109 length zero you can omit this. But in trivial algebras, it is
112 - ``field`` -- a subfield of the reals (default: ``AA``); the scalar
113 field for the algebra.
115 - ``orthonormalize`` -- boolean (default: ``True``); whether or
116 not to orthonormalize the basis. Doing so is expensive and
117 generally rules out using the rationals as your ``field``, but
118 is required for spectral decompositions.
122 sage: from mjo.eja.eja_algebra import random_eja
126 We should compute that an element subalgebra is associative even
127 if we circumvent the element method::
129 sage: set_random_seed()
130 sage: J = random_eja(field=QQ,orthonormalize=False)
131 sage: x = J.random_element()
132 sage: A = x.subalgebra_generated_by(orthonormalize=False)
133 sage: basis = tuple(b.superalgebra_element() for b in A.basis())
134 sage: J.subalgebra(basis, orthonormalize=False).is_associative()
137 Element
= FiniteDimensionalEJAElement
147 cartesian_product
=False,
155 if not field
.is_subring(RR
):
156 # Note: this does return true for the real algebraic
157 # field, the rationals, and any quadratic field where
158 # we've specified a real embedding.
159 raise ValueError("scalar field is not real")
162 # Check commutativity of the Jordan and inner-products.
163 # This has to be done before we build the multiplication
164 # and inner-product tables/matrices, because we take
165 # advantage of symmetry in the process.
166 if not all( jordan_product(bi
,bj
) == jordan_product(bj
,bi
)
169 raise ValueError("Jordan product is not commutative")
171 if not all( inner_product(bi
,bj
) == inner_product(bj
,bi
)
174 raise ValueError("inner-product is not commutative")
177 category
= MagmaticAlgebras(field
).FiniteDimensional()
178 category
= category
.WithBasis().Unital().Commutative()
181 # All zero- and one-dimensional algebras are just the real
182 # numbers with (some positive multiples of) the usual
183 # multiplication as its Jordan and inner-product.
185 if associative
is None:
186 # We should figure it out. As with check_axioms, we have to do
187 # this without the help of the _jordan_product_is_associative()
188 # method because we need to know the category before we
189 # initialize the algebra.
190 associative
= all( jordan_product(jordan_product(bi
,bj
),bk
)
192 jordan_product(bi
,jordan_product(bj
,bk
))
198 # Element subalgebras can take advantage of this.
199 category
= category
.Associative()
200 if cartesian_product
:
201 # Use join() here because otherwise we only get the
202 # "Cartesian product of..." and not the things themselves.
203 category
= category
.join([category
,
204 category
.CartesianProducts()])
206 # Call the superclass constructor so that we can use its from_vector()
207 # method to build our multiplication table.
208 CombinatorialFreeModule
.__init
__(self
,
215 # Now comes all of the hard work. We'll be constructing an
216 # ambient vector space V that our (vectorized) basis lives in,
217 # as well as a subspace W of V spanned by those (vectorized)
218 # basis elements. The W-coordinates are the coefficients that
219 # we see in things like x = 1*b1 + 2*b2.
224 degree
= len(_all2list(basis
[0]))
226 # Build an ambient space that fits our matrix basis when
227 # written out as "long vectors."
228 V
= VectorSpace(field
, degree
)
230 # The matrix that will hole the orthonormal -> unorthonormal
231 # coordinate transformation.
232 self
._deortho
_matrix
= None
235 # Save a copy of the un-orthonormalized basis for later.
236 # Convert it to ambient V (vector) coordinates while we're
237 # at it, because we'd have to do it later anyway.
238 deortho_vector_basis
= tuple( V(_all2list(b
)) for b
in basis
)
240 from mjo
.eja
.eja_utils
import gram_schmidt
241 basis
= tuple(gram_schmidt(basis
, inner_product
))
243 # Save the (possibly orthonormalized) matrix basis for
244 # later, as well as the space that its elements live in.
245 # In most cases we can deduce the matrix space, but when
246 # n == 0 (that is, there are no basis elements) we cannot.
247 self
._matrix
_basis
= basis
248 if matrix_space
is None:
249 self
._matrix
_space
= self
._matrix
_basis
[0].parent()
251 self
._matrix
_space
= matrix_space
253 # Now create the vector space for the algebra, which will have
254 # its own set of non-ambient coordinates (in terms of the
256 vector_basis
= tuple( V(_all2list(b
)) for b
in basis
)
257 W
= V
.span_of_basis( vector_basis
, check
=check_axioms
)
260 # Now "W" is the vector space of our algebra coordinates. The
261 # variables "X1", "X2",... refer to the entries of vectors in
262 # W. Thus to convert back and forth between the orthonormal
263 # coordinates and the given ones, we need to stick the original
265 U
= V
.span_of_basis( deortho_vector_basis
, check
=check_axioms
)
266 self
._deortho
_matrix
= matrix( U
.coordinate_vector(q
)
267 for q
in vector_basis
)
270 # Now we actually compute the multiplication and inner-product
271 # tables/matrices using the possibly-orthonormalized basis.
272 self
._inner
_product
_matrix
= matrix
.identity(field
, n
)
273 self
._multiplication
_table
= [ [0 for j
in range(i
+1)]
276 # Note: the Jordan and inner-products are defined in terms
277 # of the ambient basis. It's important that their arguments
278 # are in ambient coordinates as well.
281 # ortho basis w.r.t. ambient coords
285 # The jordan product returns a matrixy answer, so we
286 # have to convert it to the algebra coordinates.
287 elt
= jordan_product(q_i
, q_j
)
288 elt
= W
.coordinate_vector(V(_all2list(elt
)))
289 self
._multiplication
_table
[i
][j
] = self
.from_vector(elt
)
291 if not orthonormalize
:
292 # If we're orthonormalizing the basis with respect
293 # to an inner-product, then the inner-product
294 # matrix with respect to the resulting basis is
295 # just going to be the identity.
296 ip
= inner_product(q_i
, q_j
)
297 self
._inner
_product
_matrix
[i
,j
] = ip
298 self
._inner
_product
_matrix
[j
,i
] = ip
300 self
._inner
_product
_matrix
._cache
= {'hermitian': True}
301 self
._inner
_product
_matrix
.set_immutable()
304 if not self
._is
_jordanian
():
305 raise ValueError("Jordan identity does not hold")
306 if not self
._inner
_product
_is
_associative
():
307 raise ValueError("inner product is not associative")
310 def _coerce_map_from_base_ring(self
):
312 Disable the map from the base ring into the algebra.
314 Performing a nonsense conversion like this automatically
315 is counterpedagogical. The fallback is to try the usual
316 element constructor, which should also fail.
320 sage: from mjo.eja.eja_algebra import random_eja
324 sage: set_random_seed()
325 sage: J = random_eja()
327 Traceback (most recent call last):
329 ValueError: not an element of this algebra
335 def product_on_basis(self
, i
, j
):
337 Returns the Jordan product of the `i` and `j`th basis elements.
339 This completely defines the Jordan product on the algebra, and
340 is used direclty by our superclass machinery to implement
345 sage: from mjo.eja.eja_algebra import random_eja
349 sage: set_random_seed()
350 sage: J = random_eja()
351 sage: n = J.dimension()
354 sage: bi_bj = J.zero()*J.zero()
356 ....: i = ZZ.random_element(n)
357 ....: j = ZZ.random_element(n)
358 ....: bi = J.monomial(i)
359 ....: bj = J.monomial(j)
360 ....: bi_bj = J.product_on_basis(i,j)
365 # We only stored the lower-triangular portion of the
366 # multiplication table.
368 return self
._multiplication
_table
[i
][j
]
370 return self
._multiplication
_table
[j
][i
]
372 def inner_product(self
, x
, y
):
374 The inner product associated with this Euclidean Jordan algebra.
376 Defaults to the trace inner product, but can be overridden by
377 subclasses if they are sure that the necessary properties are
382 sage: from mjo.eja.eja_algebra import (random_eja,
384 ....: BilinearFormEJA)
388 Our inner product is "associative," which means the following for
389 a symmetric bilinear form::
391 sage: set_random_seed()
392 sage: J = random_eja()
393 sage: x,y,z = J.random_elements(3)
394 sage: (x*y).inner_product(z) == y.inner_product(x*z)
399 Ensure that this is the usual inner product for the algebras
402 sage: set_random_seed()
403 sage: J = HadamardEJA.random_instance()
404 sage: x,y = J.random_elements(2)
405 sage: actual = x.inner_product(y)
406 sage: expected = x.to_vector().inner_product(y.to_vector())
407 sage: actual == expected
410 Ensure that this is one-half of the trace inner-product in a
411 BilinearFormEJA that isn't just the reals (when ``n`` isn't
412 one). This is in Faraut and Koranyi, and also my "On the
415 sage: set_random_seed()
416 sage: J = BilinearFormEJA.random_instance()
417 sage: n = J.dimension()
418 sage: x = J.random_element()
419 sage: y = J.random_element()
420 sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2)
424 B
= self
._inner
_product
_matrix
425 return (B
*x
.to_vector()).inner_product(y
.to_vector())
428 def is_associative(self
):
430 Return whether or not this algebra's Jordan product is associative.
434 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
438 sage: J = ComplexHermitianEJA(3, field=QQ, orthonormalize=False)
439 sage: J.is_associative()
441 sage: x = sum(J.gens())
442 sage: A = x.subalgebra_generated_by(orthonormalize=False)
443 sage: A.is_associative()
447 return "Associative" in self
.category().axioms()
449 def _is_commutative(self
):
451 Whether or not this algebra's multiplication table is commutative.
453 This method should of course always return ``True``, unless
454 this algebra was constructed with ``check_axioms=False`` and
455 passed an invalid multiplication table.
457 return all( x
*y
== y
*x
for x
in self
.gens() for y
in self
.gens() )
459 def _is_jordanian(self
):
461 Whether or not this algebra's multiplication table respects the
462 Jordan identity `(x^{2})(xy) = x(x^{2}y)`.
464 We only check one arrangement of `x` and `y`, so for a
465 ``True`` result to be truly true, you should also check
466 :meth:`_is_commutative`. This method should of course always
467 return ``True``, unless this algebra was constructed with
468 ``check_axioms=False`` and passed an invalid multiplication table.
470 return all( (self
.monomial(i
)**2)*(self
.monomial(i
)*self
.monomial(j
))
472 (self
.monomial(i
))*((self
.monomial(i
)**2)*self
.monomial(j
))
473 for i
in range(self
.dimension())
474 for j
in range(self
.dimension()) )
476 def _jordan_product_is_associative(self
):
478 Return whether or not this algebra's Jordan product is
479 associative; that is, whether or not `x*(y*z) = (x*y)*z`
482 This method should agree with :meth:`is_associative` unless
483 you lied about the value of the ``associative`` parameter
484 when you constructed the algebra.
488 sage: from mjo.eja.eja_algebra import (random_eja,
489 ....: RealSymmetricEJA,
490 ....: ComplexHermitianEJA,
491 ....: QuaternionHermitianEJA)
495 sage: J = RealSymmetricEJA(4, orthonormalize=False)
496 sage: J._jordan_product_is_associative()
498 sage: x = sum(J.gens())
499 sage: A = x.subalgebra_generated_by()
500 sage: A._jordan_product_is_associative()
505 sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
506 sage: J._jordan_product_is_associative()
508 sage: x = sum(J.gens())
509 sage: A = x.subalgebra_generated_by(orthonormalize=False)
510 sage: A._jordan_product_is_associative()
515 sage: J = QuaternionHermitianEJA(2)
516 sage: J._jordan_product_is_associative()
518 sage: x = sum(J.gens())
519 sage: A = x.subalgebra_generated_by()
520 sage: A._jordan_product_is_associative()
525 The values we've presupplied to the constructors agree with
528 sage: set_random_seed()
529 sage: J = random_eja()
530 sage: J.is_associative() == J._jordan_product_is_associative()
536 # Used to check whether or not something is zero.
539 # I don't know of any examples that make this magnitude
540 # necessary because I don't know how to make an
541 # associative algebra when the element subalgebra
542 # construction is unreliable (as it is over RDF; we can't
543 # find the degree of an element because we can't compute
544 # the rank of a matrix). But even multiplication of floats
545 # is non-associative, so *some* epsilon is needed... let's
546 # just take the one from _inner_product_is_associative?
549 for i
in range(self
.dimension()):
550 for j
in range(self
.dimension()):
551 for k
in range(self
.dimension()):
555 diff
= (x
*y
)*z
- x
*(y
*z
)
557 if diff
.norm() > epsilon
:
562 def _inner_product_is_associative(self
):
564 Return whether or not this algebra's inner product `B` is
565 associative; that is, whether or not `B(xy,z) = B(x,yz)`.
567 This method should of course always return ``True``, unless
568 this algebra was constructed with ``check_axioms=False`` and
569 passed an invalid Jordan or inner-product.
573 # Used to check whether or not something is zero.
576 # This choice is sufficient to allow the construction of
577 # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True.
580 for i
in range(self
.dimension()):
581 for j
in range(self
.dimension()):
582 for k
in range(self
.dimension()):
586 diff
= (x
*y
).inner_product(z
) - x
.inner_product(y
*z
)
588 if diff
.abs() > epsilon
:
593 def _element_constructor_(self
, elt
):
595 Construct an element of this algebra from its vector or matrix
598 This gets called only after the parent element _call_ method
599 fails to find a coercion for the argument.
603 sage: from mjo.eja.eja_algebra import (random_eja,
606 ....: RealSymmetricEJA)
610 The identity in `S^n` is converted to the identity in the EJA::
612 sage: J = RealSymmetricEJA(3)
613 sage: I = matrix.identity(QQ,3)
614 sage: J(I) == J.one()
617 This skew-symmetric matrix can't be represented in the EJA::
619 sage: J = RealSymmetricEJA(3)
620 sage: A = matrix(QQ,3, lambda i,j: i-j)
622 Traceback (most recent call last):
624 ValueError: not an element of this algebra
626 Tuples work as well, provided that the matrix basis for the
627 algebra consists of them::
629 sage: J1 = HadamardEJA(3)
630 sage: J2 = RealSymmetricEJA(2)
631 sage: J = cartesian_product([J1,J2])
632 sage: J( (J1.matrix_basis()[1], J2.matrix_basis()[2]) )
637 Ensure that we can convert any element back and forth
638 faithfully between its matrix and algebra representations::
640 sage: set_random_seed()
641 sage: J = random_eja()
642 sage: x = J.random_element()
643 sage: J(x.to_matrix()) == x
646 We cannot coerce elements between algebras just because their
647 matrix representations are compatible::
649 sage: J1 = HadamardEJA(3)
650 sage: J2 = JordanSpinEJA(3)
652 Traceback (most recent call last):
654 ValueError: not an element of this algebra
656 Traceback (most recent call last):
658 ValueError: not an element of this algebra
660 msg
= "not an element of this algebra"
661 if elt
in self
.base_ring():
662 # Ensure that no base ring -> algebra coercion is performed
663 # by this method. There's some stupidity in sage that would
664 # otherwise propagate to this method; for example, sage thinks
665 # that the integer 3 belongs to the space of 2-by-2 matrices.
666 raise ValueError(msg
)
669 # Try to convert a vector into a column-matrix...
671 except (AttributeError, TypeError):
672 # and ignore failure, because we weren't really expecting
673 # a vector as an argument anyway.
676 if elt
not in self
.matrix_space():
677 raise ValueError(msg
)
679 # Thanks for nothing! Matrix spaces aren't vector spaces in
680 # Sage, so we have to figure out its matrix-basis coordinates
681 # ourselves. We use the basis space's ring instead of the
682 # element's ring because the basis space might be an algebraic
683 # closure whereas the base ring of the 3-by-3 identity matrix
684 # could be QQ instead of QQbar.
686 # And, we also have to handle Cartesian product bases (when
687 # the matrix basis consists of tuples) here. The "good news"
688 # is that we're already converting everything to long vectors,
689 # and that strategy works for tuples as well.
691 # We pass check=False because the matrix basis is "guaranteed"
692 # to be linearly independent... right? Ha ha.
694 V
= VectorSpace(self
.base_ring(), len(elt
))
695 W
= V
.span_of_basis( (V(_all2list(s
)) for s
in self
.matrix_basis()),
699 coords
= W
.coordinate_vector(V(elt
))
700 except ArithmeticError: # vector is not in free module
701 raise ValueError(msg
)
703 return self
.from_vector(coords
)
707 Return a string representation of ``self``.
711 sage: from mjo.eja.eja_algebra import JordanSpinEJA
715 Ensure that it says what we think it says::
717 sage: JordanSpinEJA(2, field=AA)
718 Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
719 sage: JordanSpinEJA(3, field=RDF)
720 Euclidean Jordan algebra of dimension 3 over Real Double Field
723 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
724 return fmt
.format(self
.dimension(), self
.base_ring())
728 def characteristic_polynomial_of(self
):
730 Return the algebra's "characteristic polynomial of" function,
731 which is itself a multivariate polynomial that, when evaluated
732 at the coordinates of some algebra element, returns that
733 element's characteristic polynomial.
735 The resulting polynomial has `n+1` variables, where `n` is the
736 dimension of this algebra. The first `n` variables correspond to
737 the coordinates of an algebra element: when evaluated at the
738 coordinates of an algebra element with respect to a certain
739 basis, the result is a univariate polynomial (in the one
740 remaining variable ``t``), namely the characteristic polynomial
745 sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA
749 The characteristic polynomial in the spin algebra is given in
750 Alizadeh, Example 11.11::
752 sage: J = JordanSpinEJA(3)
753 sage: p = J.characteristic_polynomial_of(); p
754 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
755 sage: xvec = J.one().to_vector()
759 By definition, the characteristic polynomial is a monic
760 degree-zero polynomial in a rank-zero algebra. Note that
761 Cayley-Hamilton is indeed satisfied since the polynomial
762 ``1`` evaluates to the identity element of the algebra on
765 sage: J = TrivialEJA()
766 sage: J.characteristic_polynomial_of()
773 # The list of coefficient polynomials a_0, a_1, a_2, ..., a_(r-1).
774 a
= self
._charpoly
_coefficients
()
776 # We go to a bit of trouble here to reorder the
777 # indeterminates, so that it's easier to evaluate the
778 # characteristic polynomial at x's coordinates and get back
779 # something in terms of t, which is what we want.
780 S
= PolynomialRing(self
.base_ring(),'t')
784 S
= PolynomialRing(S
, R
.variable_names())
787 return (t
**r
+ sum( a
[k
]*(t
**k
) for k
in range(r
) ))
789 def coordinate_polynomial_ring(self
):
791 The multivariate polynomial ring in which this algebra's
792 :meth:`characteristic_polynomial_of` lives.
796 sage: from mjo.eja.eja_algebra import (HadamardEJA,
797 ....: RealSymmetricEJA)
801 sage: J = HadamardEJA(2)
802 sage: J.coordinate_polynomial_ring()
803 Multivariate Polynomial Ring in X1, X2...
804 sage: J = RealSymmetricEJA(3,field=QQ,orthonormalize=False)
805 sage: J.coordinate_polynomial_ring()
806 Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6...
809 var_names
= tuple( "X%d" % z
for z
in range(1, self
.dimension()+1) )
810 return PolynomialRing(self
.base_ring(), var_names
)
812 def inner_product(self
, x
, y
):
814 The inner product associated with this Euclidean Jordan algebra.
816 Defaults to the trace inner product, but can be overridden by
817 subclasses if they are sure that the necessary properties are
822 sage: from mjo.eja.eja_algebra import (random_eja,
824 ....: BilinearFormEJA)
828 Our inner product is "associative," which means the following for
829 a symmetric bilinear form::
831 sage: set_random_seed()
832 sage: J = random_eja()
833 sage: x,y,z = J.random_elements(3)
834 sage: (x*y).inner_product(z) == y.inner_product(x*z)
839 Ensure that this is the usual inner product for the algebras
842 sage: set_random_seed()
843 sage: J = HadamardEJA.random_instance()
844 sage: x,y = J.random_elements(2)
845 sage: actual = x.inner_product(y)
846 sage: expected = x.to_vector().inner_product(y.to_vector())
847 sage: actual == expected
850 Ensure that this is one-half of the trace inner-product in a
851 BilinearFormEJA that isn't just the reals (when ``n`` isn't
852 one). This is in Faraut and Koranyi, and also my "On the
855 sage: set_random_seed()
856 sage: J = BilinearFormEJA.random_instance()
857 sage: n = J.dimension()
858 sage: x = J.random_element()
859 sage: y = J.random_element()
860 sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2)
863 B
= self
._inner
_product
_matrix
864 return (B
*x
.to_vector()).inner_product(y
.to_vector())
867 def is_trivial(self
):
869 Return whether or not this algebra is trivial.
871 A trivial algebra contains only the zero element.
875 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
880 sage: J = ComplexHermitianEJA(3)
886 sage: J = TrivialEJA()
891 return self
.dimension() == 0
894 def multiplication_table(self
):
896 Return a visual representation of this algebra's multiplication
897 table (on basis elements).
901 sage: from mjo.eja.eja_algebra import JordanSpinEJA
905 sage: J = JordanSpinEJA(4)
906 sage: J.multiplication_table()
907 +----++----+----+----+----+
908 | * || b0 | b1 | b2 | b3 |
909 +====++====+====+====+====+
910 | b0 || b0 | b1 | b2 | b3 |
911 +----++----+----+----+----+
912 | b1 || b1 | b0 | 0 | 0 |
913 +----++----+----+----+----+
914 | b2 || b2 | 0 | b0 | 0 |
915 +----++----+----+----+----+
916 | b3 || b3 | 0 | 0 | b0 |
917 +----++----+----+----+----+
921 # Prepend the header row.
922 M
= [["*"] + list(self
.gens())]
924 # And to each subsequent row, prepend an entry that belongs to
925 # the left-side "header column."
926 M
+= [ [self
.monomial(i
)] + [ self
.monomial(i
)*self
.monomial(j
)
930 return table(M
, header_row
=True, header_column
=True, frame
=True)
933 def matrix_basis(self
):
935 Return an (often more natural) representation of this algebras
936 basis as an ordered tuple of matrices.
938 Every finite-dimensional Euclidean Jordan Algebra is a, up to
939 Jordan isomorphism, a direct sum of five simple
940 algebras---four of which comprise Hermitian matrices. And the
941 last type of algebra can of course be thought of as `n`-by-`1`
942 column matrices (ambiguusly called column vectors) to avoid
943 special cases. As a result, matrices (and column vectors) are
944 a natural representation format for Euclidean Jordan algebra
947 But, when we construct an algebra from a basis of matrices,
948 those matrix representations are lost in favor of coordinate
949 vectors *with respect to* that basis. We could eventually
950 convert back if we tried hard enough, but having the original
951 representations handy is valuable enough that we simply store
952 them and return them from this method.
954 Why implement this for non-matrix algebras? Avoiding special
955 cases for the :class:`BilinearFormEJA` pays with simplicity in
956 its own right. But mainly, we would like to be able to assume
957 that elements of a :class:`CartesianProductEJA` can be displayed
958 nicely, without having to have special classes for direct sums
959 one of whose components was a matrix algebra.
963 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
964 ....: RealSymmetricEJA)
968 sage: J = RealSymmetricEJA(2)
970 Finite family {0: b0, 1: b1, 2: b2}
971 sage: J.matrix_basis()
973 [1 0] [ 0 0.7071067811865475?] [0 0]
974 [0 0], [0.7071067811865475? 0], [0 1]
979 sage: J = JordanSpinEJA(2)
981 Finite family {0: b0, 1: b1}
982 sage: J.matrix_basis()
988 return self
._matrix
_basis
991 def matrix_space(self
):
993 Return the matrix space in which this algebra's elements live, if
994 we think of them as matrices (including column vectors of the
997 "By default" this will be an `n`-by-`1` column-matrix space,
998 except when the algebra is trivial. There it's `n`-by-`n`
999 (where `n` is zero), to ensure that two elements of the matrix
1000 space (empty matrices) can be multiplied. For algebras of
1001 matrices, this returns the space in which their
1002 real embeddings live.
1006 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
1007 ....: JordanSpinEJA,
1008 ....: QuaternionHermitianEJA,
1013 By default, the matrix representation is just a column-matrix
1014 equivalent to the vector representation::
1016 sage: J = JordanSpinEJA(3)
1017 sage: J.matrix_space()
1018 Full MatrixSpace of 3 by 1 dense matrices over Algebraic
1021 The matrix representation in the trivial algebra is
1022 zero-by-zero instead of the usual `n`-by-one::
1024 sage: J = TrivialEJA()
1025 sage: J.matrix_space()
1026 Full MatrixSpace of 0 by 0 dense matrices over Algebraic
1029 The matrix space for complex/quaternion Hermitian matrix EJA
1030 is the space in which their real-embeddings live, not the
1031 original complex/quaternion matrix space::
1033 sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
1034 sage: J.matrix_space()
1035 Module of 2 by 2 matrices with entries in Algebraic Field over
1036 the scalar ring Rational Field
1037 sage: J = QuaternionHermitianEJA(1,field=QQ,orthonormalize=False)
1038 sage: J.matrix_space()
1039 Module of 1 by 1 matrices with entries in Quaternion
1040 Algebra (-1, -1) with base ring Rational Field over
1041 the scalar ring Rational Field
1044 return self
._matrix
_space
1050 Return the unit element of this algebra.
1054 sage: from mjo.eja.eja_algebra import (HadamardEJA,
1059 We can compute unit element in the Hadamard EJA::
1061 sage: J = HadamardEJA(5)
1063 b0 + b1 + b2 + b3 + b4
1065 The unit element in the Hadamard EJA is inherited in the
1066 subalgebras generated by its elements::
1068 sage: J = HadamardEJA(5)
1070 b0 + b1 + b2 + b3 + b4
1071 sage: x = sum(J.gens())
1072 sage: A = x.subalgebra_generated_by(orthonormalize=False)
1075 sage: A.one().superalgebra_element()
1076 b0 + b1 + b2 + b3 + b4
1080 The identity element acts like the identity, regardless of
1081 whether or not we orthonormalize::
1083 sage: set_random_seed()
1084 sage: J = random_eja()
1085 sage: x = J.random_element()
1086 sage: J.one()*x == x and x*J.one() == x
1088 sage: A = x.subalgebra_generated_by()
1089 sage: y = A.random_element()
1090 sage: A.one()*y == y and y*A.one() == y
1095 sage: set_random_seed()
1096 sage: J = random_eja(field=QQ, orthonormalize=False)
1097 sage: x = J.random_element()
1098 sage: J.one()*x == x and x*J.one() == x
1100 sage: A = x.subalgebra_generated_by(orthonormalize=False)
1101 sage: y = A.random_element()
1102 sage: A.one()*y == y and y*A.one() == y
1105 The matrix of the unit element's operator is the identity,
1106 regardless of the base field and whether or not we
1109 sage: set_random_seed()
1110 sage: J = random_eja()
1111 sage: actual = J.one().operator().matrix()
1112 sage: expected = matrix.identity(J.base_ring(), J.dimension())
1113 sage: actual == expected
1115 sage: x = J.random_element()
1116 sage: A = x.subalgebra_generated_by()
1117 sage: actual = A.one().operator().matrix()
1118 sage: expected = matrix.identity(A.base_ring(), A.dimension())
1119 sage: actual == expected
1124 sage: set_random_seed()
1125 sage: J = random_eja(field=QQ, orthonormalize=False)
1126 sage: actual = J.one().operator().matrix()
1127 sage: expected = matrix.identity(J.base_ring(), J.dimension())
1128 sage: actual == expected
1130 sage: x = J.random_element()
1131 sage: A = x.subalgebra_generated_by(orthonormalize=False)
1132 sage: actual = A.one().operator().matrix()
1133 sage: expected = matrix.identity(A.base_ring(), A.dimension())
1134 sage: actual == expected
1137 Ensure that the cached unit element (often precomputed by
1138 hand) agrees with the computed one::
1140 sage: set_random_seed()
1141 sage: J = random_eja()
1142 sage: cached = J.one()
1143 sage: J.one.clear_cache()
1144 sage: J.one() == cached
1149 sage: set_random_seed()
1150 sage: J = random_eja(field=QQ, orthonormalize=False)
1151 sage: cached = J.one()
1152 sage: J.one.clear_cache()
1153 sage: J.one() == cached
1157 # We can brute-force compute the matrices of the operators
1158 # that correspond to the basis elements of this algebra.
1159 # If some linear combination of those basis elements is the
1160 # algebra identity, then the same linear combination of
1161 # their matrices has to be the identity matrix.
1163 # Of course, matrices aren't vectors in sage, so we have to
1164 # appeal to the "long vectors" isometry.
1165 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
1167 # Now we use basic linear algebra to find the coefficients,
1168 # of the matrices-as-vectors-linear-combination, which should
1169 # work for the original algebra basis too.
1170 A
= matrix(self
.base_ring(), oper_vecs
)
1172 # We used the isometry on the left-hand side already, but we
1173 # still need to do it for the right-hand side. Recall that we
1174 # wanted something that summed to the identity matrix.
1175 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
1177 # Now if there's an identity element in the algebra, this
1178 # should work. We solve on the left to avoid having to
1179 # transpose the matrix "A".
1180 return self
.from_vector(A
.solve_left(b
))
1183 def peirce_decomposition(self
, c
):
1185 The Peirce decomposition of this algebra relative to the
1188 In the future, this can be extended to a complete system of
1189 orthogonal idempotents.
1193 - ``c`` -- an idempotent of this algebra.
1197 A triple (J0, J5, J1) containing two subalgebras and one subspace
1200 - ``J0`` -- the algebra on the eigenspace of ``c.operator()``
1201 corresponding to the eigenvalue zero.
1203 - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()``
1204 corresponding to the eigenvalue one-half.
1206 - ``J1`` -- the algebra on the eigenspace of ``c.operator()``
1207 corresponding to the eigenvalue one.
1209 These are the only possible eigenspaces for that operator, and this
1210 algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are
1211 orthogonal, and are subalgebras of this algebra with the appropriate
1216 sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA
1220 The canonical example comes from the symmetric matrices, which
1221 decompose into diagonal and off-diagonal parts::
1223 sage: J = RealSymmetricEJA(3)
1224 sage: C = matrix(QQ, [ [1,0,0],
1228 sage: J0,J5,J1 = J.peirce_decomposition(c)
1230 Euclidean Jordan algebra of dimension 1...
1232 Vector space of degree 6 and dimension 2...
1234 Euclidean Jordan algebra of dimension 3...
1235 sage: J0.one().to_matrix()
1239 sage: orig_df = AA.options.display_format
1240 sage: AA.options.display_format = 'radical'
1241 sage: J.from_vector(J5.basis()[0]).to_matrix()
1245 sage: J.from_vector(J5.basis()[1]).to_matrix()
1249 sage: AA.options.display_format = orig_df
1250 sage: J1.one().to_matrix()
1257 Every algebra decomposes trivially with respect to its identity
1260 sage: set_random_seed()
1261 sage: J = random_eja()
1262 sage: J0,J5,J1 = J.peirce_decomposition(J.one())
1263 sage: J0.dimension() == 0 and J5.dimension() == 0
1265 sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
1268 The decomposition is into eigenspaces, and its components are
1269 therefore necessarily orthogonal. Moreover, the identity
1270 elements in the two subalgebras are the projections onto their
1271 respective subspaces of the superalgebra's identity element::
1273 sage: set_random_seed()
1274 sage: J = random_eja()
1275 sage: x = J.random_element()
1276 sage: if not J.is_trivial():
1277 ....: while x.is_nilpotent():
1278 ....: x = J.random_element()
1279 sage: c = x.subalgebra_idempotent()
1280 sage: J0,J5,J1 = J.peirce_decomposition(c)
1282 sage: for (w,y,z) in zip(J0.basis(), J5.basis(), J1.basis()):
1283 ....: w = w.superalgebra_element()
1284 ....: y = J.from_vector(y)
1285 ....: z = z.superalgebra_element()
1286 ....: ipsum += w.inner_product(y).abs()
1287 ....: ipsum += w.inner_product(z).abs()
1288 ....: ipsum += y.inner_product(z).abs()
1291 sage: J1(c) == J1.one()
1293 sage: J0(J.one() - c) == J0.one()
1297 if not c
.is_idempotent():
1298 raise ValueError("element is not idempotent: %s" % c
)
1300 # Default these to what they should be if they turn out to be
1301 # trivial, because eigenspaces_left() won't return eigenvalues
1302 # corresponding to trivial spaces (e.g. it returns only the
1303 # eigenspace corresponding to lambda=1 if you take the
1304 # decomposition relative to the identity element).
1305 trivial
= self
.subalgebra(())
1306 J0
= trivial
# eigenvalue zero
1307 J5
= VectorSpace(self
.base_ring(), 0) # eigenvalue one-half
1308 J1
= trivial
# eigenvalue one
1310 for (eigval
, eigspace
) in c
.operator().matrix().right_eigenspaces():
1311 if eigval
== ~
(self
.base_ring()(2)):
1314 gens
= tuple( self
.from_vector(b
) for b
in eigspace
.basis() )
1315 subalg
= self
.subalgebra(gens
, check_axioms
=False)
1321 raise ValueError("unexpected eigenvalue: %s" % eigval
)
1326 def random_element(self
, thorough
=False):
1328 Return a random element of this algebra.
1330 Our algebra superclass method only returns a linear
1331 combination of at most two basis elements. We instead
1332 want the vector space "random element" method that
1333 returns a more diverse selection.
1337 - ``thorough`` -- (boolean; default False) whether or not we
1338 should generate irrational coefficients for the random
1339 element when our base ring is irrational; this slows the
1340 algebra operations to a crawl, but any truly random method
1344 # For a general base ring... maybe we can trust this to do the
1345 # right thing? Unlikely, but.
1346 V
= self
.vector_space()
1347 v
= V
.random_element()
1349 if self
.base_ring() is AA
:
1350 # The "random element" method of the algebraic reals is
1351 # stupid at the moment, and only returns integers between
1352 # -2 and 2, inclusive:
1354 # https://trac.sagemath.org/ticket/30875
1356 # Instead, we implement our own "random vector" method,
1357 # and then coerce that into the algebra. We use the vector
1358 # space degree here instead of the dimension because a
1359 # subalgebra could (for example) be spanned by only two
1360 # vectors, each with five coordinates. We need to
1361 # generate all five coordinates.
1363 v
*= QQbar
.random_element().real()
1365 v
*= QQ
.random_element()
1367 return self
.from_vector(V
.coordinate_vector(v
))
1369 def random_elements(self
, count
, thorough
=False):
1371 Return ``count`` random elements as a tuple.
1375 - ``thorough`` -- (boolean; default False) whether or not we
1376 should generate irrational coefficients for the random
1377 elements when our base ring is irrational; this slows the
1378 algebra operations to a crawl, but any truly random method
1383 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1387 sage: J = JordanSpinEJA(3)
1388 sage: x,y,z = J.random_elements(3)
1389 sage: all( [ x in J, y in J, z in J ])
1391 sage: len( J.random_elements(10) ) == 10
1395 return tuple( self
.random_element(thorough
)
1396 for idx
in range(count
) )
1400 def _charpoly_coefficients(self
):
1402 The `r` polynomial coefficients of the "characteristic polynomial
1407 sage: from mjo.eja.eja_algebra import random_eja
1411 The theory shows that these are all homogeneous polynomials of
1414 sage: set_random_seed()
1415 sage: J = random_eja()
1416 sage: all(p.is_homogeneous() for p in J._charpoly_coefficients())
1420 n
= self
.dimension()
1421 R
= self
.coordinate_polynomial_ring()
1423 F
= R
.fraction_field()
1426 # From a result in my book, these are the entries of the
1427 # basis representation of L_x.
1428 return sum( vars[k
]*self
.monomial(k
).operator().matrix()[i
,j
]
1431 L_x
= matrix(F
, n
, n
, L_x_i_j
)
1434 if self
.rank
.is_in_cache():
1436 # There's no need to pad the system with redundant
1437 # columns if we *know* they'll be redundant.
1440 # Compute an extra power in case the rank is equal to
1441 # the dimension (otherwise, we would stop at x^(r-1)).
1442 x_powers
= [ (L_x
**k
)*self
.one().to_vector()
1443 for k
in range(n
+1) ]
1444 A
= matrix
.column(F
, x_powers
[:n
])
1445 AE
= A
.extended_echelon_form()
1452 # The theory says that only the first "r" coefficients are
1453 # nonzero, and they actually live in the original polynomial
1454 # ring and not the fraction field. We negate them because in
1455 # the actual characteristic polynomial, they get moved to the
1456 # other side where x^r lives. We don't bother to trim A_rref
1457 # down to a square matrix and solve the resulting system,
1458 # because the upper-left r-by-r portion of A_rref is
1459 # guaranteed to be the identity matrix, so e.g.
1461 # A_rref.solve_right(Y)
1463 # would just be returning Y.
1464 return (-E
*b
)[:r
].change_ring(R
)
1469 Return the rank of this EJA.
1471 This is a cached method because we know the rank a priori for
1472 all of the algebras we can construct. Thus we can avoid the
1473 expensive ``_charpoly_coefficients()`` call unless we truly
1474 need to compute the whole characteristic polynomial.
1478 sage: from mjo.eja.eja_algebra import (HadamardEJA,
1479 ....: JordanSpinEJA,
1480 ....: RealSymmetricEJA,
1481 ....: ComplexHermitianEJA,
1482 ....: QuaternionHermitianEJA,
1487 The rank of the Jordan spin algebra is always two::
1489 sage: JordanSpinEJA(2).rank()
1491 sage: JordanSpinEJA(3).rank()
1493 sage: JordanSpinEJA(4).rank()
1496 The rank of the `n`-by-`n` Hermitian real, complex, or
1497 quaternion matrices is `n`::
1499 sage: RealSymmetricEJA(4).rank()
1501 sage: ComplexHermitianEJA(3).rank()
1503 sage: QuaternionHermitianEJA(2).rank()
1508 Ensure that every EJA that we know how to construct has a
1509 positive integer rank, unless the algebra is trivial in
1510 which case its rank will be zero::
1512 sage: set_random_seed()
1513 sage: J = random_eja()
1517 sage: r > 0 or (r == 0 and J.is_trivial())
1520 Ensure that computing the rank actually works, since the ranks
1521 of all simple algebras are known and will be cached by default::
1523 sage: set_random_seed() # long time
1524 sage: J = random_eja() # long time
1525 sage: cached = J.rank() # long time
1526 sage: J.rank.clear_cache() # long time
1527 sage: J.rank() == cached # long time
1531 return len(self
._charpoly
_coefficients
())
1534 def subalgebra(self
, basis
, **kwargs
):
1536 Create a subalgebra of this algebra from the given basis.
1538 from mjo
.eja
.eja_subalgebra
import FiniteDimensionalEJASubalgebra
1539 return FiniteDimensionalEJASubalgebra(self
, basis
, **kwargs
)
1542 def vector_space(self
):
1544 Return the vector space that underlies this algebra.
1548 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1552 sage: J = RealSymmetricEJA(2)
1553 sage: J.vector_space()
1554 Vector space of dimension 3 over...
1557 return self
.zero().to_vector().parent().ambient_vector_space()
1561 class RationalBasisEJA(FiniteDimensionalEJA
):
1563 Algebras whose supplied basis elements have all rational entries.
1567 sage: from mjo.eja.eja_algebra import BilinearFormEJA
1571 The supplied basis is orthonormalized by default::
1573 sage: B = matrix(QQ, [[1, 0, 0], [0, 25, -32], [0, -32, 41]])
1574 sage: J = BilinearFormEJA(B)
1575 sage: J.matrix_basis()
1592 # Abuse the check_field parameter to check that the entries of
1593 # out basis (in ambient coordinates) are in the field QQ.
1594 # Use _all2list to get the vector coordinates of octonion
1595 # entries and not the octonions themselves (which are not
1597 if not all( all(b_i
in QQ
for b_i
in _all2list(b
))
1599 raise TypeError("basis not rational")
1601 super().__init
__(basis
,
1605 check_field
=check_field
,
1608 self
._rational
_algebra
= None
1610 # There's no point in constructing the extra algebra if this
1611 # one is already rational.
1613 # Note: the same Jordan and inner-products work here,
1614 # because they are necessarily defined with respect to
1615 # ambient coordinates and not any particular basis.
1616 self
._rational
_algebra
= FiniteDimensionalEJA(
1621 matrix_space
=self
.matrix_space(),
1622 associative
=self
.is_associative(),
1623 orthonormalize
=False,
1628 def _charpoly_coefficients(self
):
1632 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
1633 ....: JordanSpinEJA)
1637 The base ring of the resulting polynomial coefficients is what
1638 it should be, and not the rationals (unless the algebra was
1639 already over the rationals)::
1641 sage: J = JordanSpinEJA(3)
1642 sage: J._charpoly_coefficients()
1643 (X1^2 - X2^2 - X3^2, -2*X1)
1644 sage: a0 = J._charpoly_coefficients()[0]
1646 Algebraic Real Field
1647 sage: a0.base_ring()
1648 Algebraic Real Field
1651 if self
._rational
_algebra
is None:
1652 # There's no need to construct *another* algebra over the
1653 # rationals if this one is already over the
1654 # rationals. Likewise, if we never orthonormalized our
1655 # basis, we might as well just use the given one.
1656 return super()._charpoly
_coefficients
()
1658 # Do the computation over the rationals. The answer will be
1659 # the same, because all we've done is a change of basis.
1660 # Then, change back from QQ to our real base ring
1661 a
= ( a_i
.change_ring(self
.base_ring())
1662 for a_i
in self
._rational
_algebra
._charpoly
_coefficients
() )
1664 if self
._deortho
_matrix
is None:
1665 # This can happen if our base ring was, say, AA and we
1666 # chose not to (or didn't need to) orthonormalize. It's
1667 # still faster to do the computations over QQ even if
1668 # the numbers in the boxes stay the same.
1671 # Otherwise, convert the coordinate variables back to the
1672 # deorthonormalized ones.
1673 R
= self
.coordinate_polynomial_ring()
1674 from sage
.modules
.free_module_element
import vector
1675 X
= vector(R
, R
.gens())
1676 BX
= self
._deortho
_matrix
*X
1678 subs_dict
= { X[i]: BX[i] for i in range(len(X)) }
1679 return tuple( a_i
.subs(subs_dict
) for a_i
in a
)
1681 class ConcreteEJA(FiniteDimensionalEJA
):
1683 A class for the Euclidean Jordan algebras that we know by name.
1685 These are the Jordan algebras whose basis, multiplication table,
1686 rank, and so on are known a priori. More to the point, they are
1687 the Euclidean Jordan algebras for which we are able to conjure up
1688 a "random instance."
1692 sage: from mjo.eja.eja_algebra import ConcreteEJA
1696 Our basis is normalized with respect to the algebra's inner
1697 product, unless we specify otherwise::
1699 sage: set_random_seed()
1700 sage: J = ConcreteEJA.random_instance()
1701 sage: all( b.norm() == 1 for b in J.gens() )
1704 Since our basis is orthonormal with respect to the algebra's inner
1705 product, and since we know that this algebra is an EJA, any
1706 left-multiplication operator's matrix will be symmetric because
1707 natural->EJA basis representation is an isometry and within the
1708 EJA the operator is self-adjoint by the Jordan axiom::
1710 sage: set_random_seed()
1711 sage: J = ConcreteEJA.random_instance()
1712 sage: x = J.random_element()
1713 sage: x.operator().is_self_adjoint()
1718 def _max_random_instance_size():
1720 Return an integer "size" that is an upper bound on the size of
1721 this algebra when it is used in a random test
1722 case. Unfortunately, the term "size" is ambiguous -- when
1723 dealing with `R^n` under either the Hadamard or Jordan spin
1724 product, the "size" refers to the dimension `n`. When dealing
1725 with a matrix algebra (real symmetric or complex/quaternion
1726 Hermitian), it refers to the size of the matrix, which is far
1727 less than the dimension of the underlying vector space.
1729 This method must be implemented in each subclass.
1731 raise NotImplementedError
1734 def random_instance(cls
, *args
, **kwargs
):
1736 Return a random instance of this type of algebra.
1738 This method should be implemented in each subclass.
1740 from sage
.misc
.prandom
import choice
1741 eja_class
= choice(cls
.__subclasses
__())
1743 # These all bubble up to the RationalBasisEJA superclass
1744 # constructor, so any (kw)args valid there are also valid
1746 return eja_class
.random_instance(*args
, **kwargs
)
1749 class MatrixEJA(FiniteDimensionalEJA
):
1751 def _denormalized_basis(A
):
1753 Returns a basis for the space of complex Hermitian n-by-n matrices.
1755 Why do we embed these? Basically, because all of numerical linear
1756 algebra assumes that you're working with vectors consisting of `n`
1757 entries from a field and scalars from the same field. There's no way
1758 to tell SageMath that (for example) the vectors contain complex
1759 numbers, while the scalar field is real.
1763 sage: from mjo.hurwitz import (ComplexMatrixAlgebra,
1764 ....: QuaternionMatrixAlgebra,
1765 ....: OctonionMatrixAlgebra)
1766 sage: from mjo.eja.eja_algebra import MatrixEJA
1770 sage: set_random_seed()
1771 sage: n = ZZ.random_element(1,5)
1772 sage: A = MatrixSpace(QQ, n)
1773 sage: B = MatrixEJA._denormalized_basis(A)
1774 sage: all( M.is_hermitian() for M in B)
1779 sage: set_random_seed()
1780 sage: n = ZZ.random_element(1,5)
1781 sage: A = ComplexMatrixAlgebra(n, scalars=QQ)
1782 sage: B = MatrixEJA._denormalized_basis(A)
1783 sage: all( M.is_hermitian() for M in B)
1788 sage: set_random_seed()
1789 sage: n = ZZ.random_element(1,5)
1790 sage: A = QuaternionMatrixAlgebra(n, scalars=QQ)
1791 sage: B = MatrixEJA._denormalized_basis(A)
1792 sage: all( M.is_hermitian() for M in B )
1797 sage: set_random_seed()
1798 sage: n = ZZ.random_element(1,5)
1799 sage: A = OctonionMatrixAlgebra(n, scalars=QQ)
1800 sage: B = MatrixEJA._denormalized_basis(A)
1801 sage: all( M.is_hermitian() for M in B )
1805 # These work for real MatrixSpace, whose monomials only have
1806 # two coordinates (because the last one would always be "1").
1807 es
= A
.base_ring().gens()
1808 gen
= lambda A
,m
: A
.monomial(m
[:2])
1810 if hasattr(A
, 'entry_algebra_gens'):
1811 # We've got a MatrixAlgebra, and its monomials will have
1812 # three coordinates.
1813 es
= A
.entry_algebra_gens()
1814 gen
= lambda A
,m
: A
.monomial(m
)
1817 for i
in range(A
.nrows()):
1818 for j
in range(i
+1):
1820 E_ii
= gen(A
, (i
,j
,es
[0]))
1824 E_ij
= gen(A
, (i
,j
,e
))
1825 E_ij
+= E_ij
.conjugate_transpose()
1828 return tuple( basis
)
1831 def jordan_product(X
,Y
):
1832 return (X
*Y
+ Y
*X
)/2
1835 def trace_inner_product(X
,Y
):
1837 A trace inner-product for matrices that aren't embedded in the
1838 reals. It takes MATRICES as arguments, not EJA elements.
1842 sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
1843 ....: ComplexHermitianEJA,
1844 ....: QuaternionHermitianEJA,
1845 ....: OctonionHermitianEJA)
1849 sage: J = RealSymmetricEJA(2,field=QQ,orthonormalize=False)
1850 sage: I = J.one().to_matrix()
1851 sage: J.trace_inner_product(I, -I)
1856 sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
1857 sage: I = J.one().to_matrix()
1858 sage: J.trace_inner_product(I, -I)
1863 sage: J = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
1864 sage: I = J.one().to_matrix()
1865 sage: J.trace_inner_product(I, -I)
1870 sage: J = OctonionHermitianEJA(2,field=QQ,orthonormalize=False)
1871 sage: I = J.one().to_matrix()
1872 sage: J.trace_inner_product(I, -I)
1877 if hasattr(tr
, 'coefficient'):
1878 # Works for octonions, and has to come first because they
1879 # also have a "real()" method that doesn't return an
1880 # element of the scalar ring.
1881 return tr
.coefficient(0)
1882 elif hasattr(tr
, 'coefficient_tuple'):
1883 # Works for quaternions.
1884 return tr
.coefficient_tuple()[0]
1886 # Works for real and complex numbers.
1890 def __init__(self
, matrix_space
, **kwargs
):
1891 # We know this is a valid EJA, but will double-check
1892 # if the user passes check_axioms=True.
1893 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
1896 super().__init
__(self
._denormalized
_basis
(matrix_space
),
1897 self
.jordan_product
,
1898 self
.trace_inner_product
,
1899 field
=matrix_space
.base_ring(),
1900 matrix_space
=matrix_space
,
1903 self
.rank
.set_cache(matrix_space
.nrows())
1904 self
.one
.set_cache( self(matrix_space
.one()) )
1906 class RealSymmetricEJA(MatrixEJA
, RationalBasisEJA
, ConcreteEJA
):
1908 The rank-n simple EJA consisting of real symmetric n-by-n
1909 matrices, the usual symmetric Jordan product, and the trace inner
1910 product. It has dimension `(n^2 + n)/2` over the reals.
1914 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1918 sage: J = RealSymmetricEJA(2)
1919 sage: b0, b1, b2 = J.gens()
1927 In theory, our "field" can be any subfield of the reals::
1929 sage: RealSymmetricEJA(2, field=RDF, check_axioms=True)
1930 Euclidean Jordan algebra of dimension 3 over Real Double Field
1931 sage: RealSymmetricEJA(2, field=RR, check_axioms=True)
1932 Euclidean Jordan algebra of dimension 3 over Real Field with
1933 53 bits of precision
1937 The dimension of this algebra is `(n^2 + n) / 2`::
1939 sage: set_random_seed()
1940 sage: n_max = RealSymmetricEJA._max_random_instance_size()
1941 sage: n = ZZ.random_element(1, n_max)
1942 sage: J = RealSymmetricEJA(n)
1943 sage: J.dimension() == (n^2 + n)/2
1946 The Jordan multiplication is what we think it is::
1948 sage: set_random_seed()
1949 sage: J = RealSymmetricEJA.random_instance()
1950 sage: x,y = J.random_elements(2)
1951 sage: actual = (x*y).to_matrix()
1952 sage: X = x.to_matrix()
1953 sage: Y = y.to_matrix()
1954 sage: expected = (X*Y + Y*X)/2
1955 sage: actual == expected
1957 sage: J(expected) == x*y
1960 We can change the generator prefix::
1962 sage: RealSymmetricEJA(3, prefix='q').gens()
1963 (q0, q1, q2, q3, q4, q5)
1965 We can construct the (trivial) algebra of rank zero::
1967 sage: RealSymmetricEJA(0)
1968 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
1972 def _max_random_instance_size():
1973 return 4 # Dimension 10
1976 def random_instance(cls
, **kwargs
):
1978 Return a random instance of this type of algebra.
1980 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
1981 return cls(n
, **kwargs
)
1983 def __init__(self
, n
, field
=AA
, **kwargs
):
1984 # We know this is a valid EJA, but will double-check
1985 # if the user passes check_axioms=True.
1986 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
1988 A
= MatrixSpace(field
, n
)
1989 super().__init
__(A
, **kwargs
)
1993 class ComplexHermitianEJA(MatrixEJA
, RationalBasisEJA
, ConcreteEJA
):
1995 The rank-n simple EJA consisting of complex Hermitian n-by-n
1996 matrices over the real numbers, the usual symmetric Jordan product,
1997 and the real-part-of-trace inner product. It has dimension `n^2` over
2002 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
2006 In theory, our "field" can be any subfield of the reals, but we
2007 can't use inexact real fields at the moment because SageMath
2008 doesn't know how to convert their elements into complex numbers,
2009 or even into algebraic reals::
2012 Traceback (most recent call last):
2014 TypeError: Illegal initializer for algebraic number
2016 Traceback (most recent call last):
2018 TypeError: Illegal initializer for algebraic number
2020 This causes the following error when we try to scale a matrix of
2021 complex numbers by an inexact real number::
2023 sage: ComplexHermitianEJA(2,field=RR)
2024 Traceback (most recent call last):
2026 TypeError: Unable to coerce entries (=(1.00000000000000,
2027 -0.000000000000000)) to coefficients in Algebraic Real Field
2031 The dimension of this algebra is `n^2`::
2033 sage: set_random_seed()
2034 sage: n_max = ComplexHermitianEJA._max_random_instance_size()
2035 sage: n = ZZ.random_element(1, n_max)
2036 sage: J = ComplexHermitianEJA(n)
2037 sage: J.dimension() == n^2
2040 The Jordan multiplication is what we think it is::
2042 sage: set_random_seed()
2043 sage: J = ComplexHermitianEJA.random_instance()
2044 sage: x,y = J.random_elements(2)
2045 sage: actual = (x*y).to_matrix()
2046 sage: X = x.to_matrix()
2047 sage: Y = y.to_matrix()
2048 sage: expected = (X*Y + Y*X)/2
2049 sage: actual == expected
2051 sage: J(expected) == x*y
2054 We can change the generator prefix::
2056 sage: ComplexHermitianEJA(2, prefix='z').gens()
2059 We can construct the (trivial) algebra of rank zero::
2061 sage: ComplexHermitianEJA(0)
2062 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
2064 def __init__(self
, n
, field
=AA
, **kwargs
):
2065 # We know this is a valid EJA, but will double-check
2066 # if the user passes check_axioms=True.
2067 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2069 from mjo
.hurwitz
import ComplexMatrixAlgebra
2070 A
= ComplexMatrixAlgebra(n
, scalars
=field
)
2071 super().__init
__(A
, **kwargs
)
2075 def _max_random_instance_size():
2076 return 3 # Dimension 9
2079 def random_instance(cls
, **kwargs
):
2081 Return a random instance of this type of algebra.
2083 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2084 return cls(n
, **kwargs
)
2087 class QuaternionHermitianEJA(MatrixEJA
, RationalBasisEJA
, ConcreteEJA
):
2089 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
2090 matrices, the usual symmetric Jordan product, and the
2091 real-part-of-trace inner product. It has dimension `2n^2 - n` over
2096 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
2100 In theory, our "field" can be any subfield of the reals::
2102 sage: QuaternionHermitianEJA(2, field=RDF, check_axioms=True)
2103 Euclidean Jordan algebra of dimension 6 over Real Double Field
2104 sage: QuaternionHermitianEJA(2, field=RR, check_axioms=True)
2105 Euclidean Jordan algebra of dimension 6 over Real Field with
2106 53 bits of precision
2110 The dimension of this algebra is `2*n^2 - n`::
2112 sage: set_random_seed()
2113 sage: n_max = QuaternionHermitianEJA._max_random_instance_size()
2114 sage: n = ZZ.random_element(1, n_max)
2115 sage: J = QuaternionHermitianEJA(n)
2116 sage: J.dimension() == 2*(n^2) - n
2119 The Jordan multiplication is what we think it is::
2121 sage: set_random_seed()
2122 sage: J = QuaternionHermitianEJA.random_instance()
2123 sage: x,y = J.random_elements(2)
2124 sage: actual = (x*y).to_matrix()
2125 sage: X = x.to_matrix()
2126 sage: Y = y.to_matrix()
2127 sage: expected = (X*Y + Y*X)/2
2128 sage: actual == expected
2130 sage: J(expected) == x*y
2133 We can change the generator prefix::
2135 sage: QuaternionHermitianEJA(2, prefix='a').gens()
2136 (a0, a1, a2, a3, a4, a5)
2138 We can construct the (trivial) algebra of rank zero::
2140 sage: QuaternionHermitianEJA(0)
2141 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
2144 def __init__(self
, n
, field
=AA
, **kwargs
):
2145 # We know this is a valid EJA, but will double-check
2146 # if the user passes check_axioms=True.
2147 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2149 from mjo
.hurwitz
import QuaternionMatrixAlgebra
2150 A
= QuaternionMatrixAlgebra(n
, scalars
=field
)
2151 super().__init
__(A
, **kwargs
)
2155 def _max_random_instance_size():
2157 The maximum rank of a random QuaternionHermitianEJA.
2159 return 2 # Dimension 6
2162 def random_instance(cls
, **kwargs
):
2164 Return a random instance of this type of algebra.
2166 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2167 return cls(n
, **kwargs
)
2169 class OctonionHermitianEJA(MatrixEJA
, RationalBasisEJA
, ConcreteEJA
):
2173 sage: from mjo.eja.eja_algebra import (FiniteDimensionalEJA,
2174 ....: OctonionHermitianEJA)
2175 sage: from mjo.hurwitz import Octonions, OctonionMatrixAlgebra
2179 The 3-by-3 algebra satisfies the axioms of an EJA::
2181 sage: OctonionHermitianEJA(3, # long time
2182 ....: field=QQ, # long time
2183 ....: orthonormalize=False, # long time
2184 ....: check_axioms=True) # long time
2185 Euclidean Jordan algebra of dimension 27 over Rational Field
2187 After a change-of-basis, the 2-by-2 algebra has the same
2188 multiplication table as the ten-dimensional Jordan spin algebra::
2190 sage: A = OctonionMatrixAlgebra(2,Octonions(QQ),QQ)
2191 sage: b = OctonionHermitianEJA._denormalized_basis(A)
2192 sage: basis = (b[0] + b[9],) + b[1:9] + (b[0] - b[9],)
2193 sage: jp = OctonionHermitianEJA.jordan_product
2194 sage: ip = OctonionHermitianEJA.trace_inner_product
2195 sage: J = FiniteDimensionalEJA(basis,
2199 ....: orthonormalize=False)
2200 sage: J.multiplication_table()
2201 +----++----+----+----+----+----+----+----+----+----+----+
2202 | * || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
2203 +====++====+====+====+====+====+====+====+====+====+====+
2204 | b0 || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
2205 +----++----+----+----+----+----+----+----+----+----+----+
2206 | b1 || b1 | b0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
2207 +----++----+----+----+----+----+----+----+----+----+----+
2208 | b2 || b2 | 0 | b0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
2209 +----++----+----+----+----+----+----+----+----+----+----+
2210 | b3 || b3 | 0 | 0 | b0 | 0 | 0 | 0 | 0 | 0 | 0 |
2211 +----++----+----+----+----+----+----+----+----+----+----+
2212 | b4 || b4 | 0 | 0 | 0 | b0 | 0 | 0 | 0 | 0 | 0 |
2213 +----++----+----+----+----+----+----+----+----+----+----+
2214 | b5 || b5 | 0 | 0 | 0 | 0 | b0 | 0 | 0 | 0 | 0 |
2215 +----++----+----+----+----+----+----+----+----+----+----+
2216 | b6 || b6 | 0 | 0 | 0 | 0 | 0 | b0 | 0 | 0 | 0 |
2217 +----++----+----+----+----+----+----+----+----+----+----+
2218 | b7 || b7 | 0 | 0 | 0 | 0 | 0 | 0 | b0 | 0 | 0 |
2219 +----++----+----+----+----+----+----+----+----+----+----+
2220 | b8 || b8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | b0 | 0 |
2221 +----++----+----+----+----+----+----+----+----+----+----+
2222 | b9 || b9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | b0 |
2223 +----++----+----+----+----+----+----+----+----+----+----+
2227 We can actually construct the 27-dimensional Albert algebra,
2228 and we get the right unit element if we recompute it::
2230 sage: J = OctonionHermitianEJA(3, # long time
2231 ....: field=QQ, # long time
2232 ....: orthonormalize=False) # long time
2233 sage: J.one.clear_cache() # long time
2234 sage: J.one() # long time
2236 sage: J.one().to_matrix() # long time
2245 The 2-by-2 algebra is isomorphic to the ten-dimensional Jordan
2246 spin algebra, but just to be sure, we recompute its rank::
2248 sage: J = OctonionHermitianEJA(2, # long time
2249 ....: field=QQ, # long time
2250 ....: orthonormalize=False) # long time
2251 sage: J.rank.clear_cache() # long time
2252 sage: J.rank() # long time
2257 def _max_random_instance_size():
2259 The maximum rank of a random QuaternionHermitianEJA.
2261 return 1 # Dimension 1
2264 def random_instance(cls
, **kwargs
):
2266 Return a random instance of this type of algebra.
2268 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2269 return cls(n
, **kwargs
)
2271 def __init__(self
, n
, field
=AA
, **kwargs
):
2273 # Otherwise we don't get an EJA.
2274 raise ValueError("n cannot exceed 3")
2276 # We know this is a valid EJA, but will double-check
2277 # if the user passes check_axioms=True.
2278 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2280 from mjo
.hurwitz
import OctonionMatrixAlgebra
2281 A
= OctonionMatrixAlgebra(n
, scalars
=field
)
2282 super().__init
__(A
, **kwargs
)
2285 from mjo
.eja
.eja_cache
import albert_eja_coeffs
2286 a
= albert_eja_coeffs(self
.coordinate_polynomial_ring())
2287 if self
._rational
_algebra
is None:
2288 self
._charpoly
_coefficients
.set_cache(a
)
2290 self
._rational
_algebra
._charpoly
_coefficients
.set_cache(a
)
2293 class AlbertEJA(OctonionHermitianEJA
):
2295 The Albert algebra is the algebra of three-by-three Hermitian
2296 matrices whose entries are octonions.
2300 sage: from mjo.eja.eja_algebra import AlbertEJA
2304 sage: AlbertEJA(field=QQ, orthonormalize=False)
2305 Euclidean Jordan algebra of dimension 27 over Rational Field
2306 sage: AlbertEJA() # long time
2307 Euclidean Jordan algebra of dimension 27 over Algebraic Real Field
2310 def __init__(self
, *args
, **kwargs
):
2311 super().__init
__(3, *args
, **kwargs
)
2314 class HadamardEJA(RationalBasisEJA
, ConcreteEJA
):
2316 Return the Euclidean Jordan algebra on `R^n` with the Hadamard
2317 (pointwise real-number multiplication) Jordan product and the
2318 usual inner-product.
2320 This is nothing more than the Cartesian product of ``n`` copies of
2321 the one-dimensional Jordan spin algebra, and is the most common
2322 example of a non-simple Euclidean Jordan algebra.
2326 sage: from mjo.eja.eja_algebra import HadamardEJA
2330 This multiplication table can be verified by hand::
2332 sage: J = HadamardEJA(3)
2333 sage: b0,b1,b2 = J.gens()
2349 We can change the generator prefix::
2351 sage: HadamardEJA(3, prefix='r').gens()
2354 def __init__(self
, n
, field
=AA
, **kwargs
):
2355 MS
= MatrixSpace(field
, n
, 1)
2358 jordan_product
= lambda x
,y
: x
2359 inner_product
= lambda x
,y
: x
2361 def jordan_product(x
,y
):
2362 return MS( xi
*yi
for (xi
,yi
) in zip(x
,y
) )
2364 def inner_product(x
,y
):
2367 # New defaults for keyword arguments. Don't orthonormalize
2368 # because our basis is already orthonormal with respect to our
2369 # inner-product. Don't check the axioms, because we know this
2370 # is a valid EJA... but do double-check if the user passes
2371 # check_axioms=True. Note: we DON'T override the "check_field"
2372 # default here, because the user can pass in a field!
2373 if "orthonormalize" not in kwargs
: kwargs
["orthonormalize"] = False
2374 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2376 column_basis
= tuple( MS(b
) for b
in FreeModule(field
, n
).basis() )
2377 super().__init
__(column_basis
,
2384 self
.rank
.set_cache(n
)
2386 self
.one
.set_cache( self
.sum(self
.gens()) )
2389 def _max_random_instance_size():
2391 The maximum dimension of a random HadamardEJA.
2396 def random_instance(cls
, **kwargs
):
2398 Return a random instance of this type of algebra.
2400 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2401 return cls(n
, **kwargs
)
2404 class BilinearFormEJA(RationalBasisEJA
, ConcreteEJA
):
2406 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2407 with the half-trace inner product and jordan product ``x*y =
2408 (<Bx,y>,y_bar>, x0*y_bar + y0*x_bar)`` where `B = 1 \times B22` is
2409 a symmetric positive-definite "bilinear form" matrix. Its
2410 dimension is the size of `B`, and it has rank two in dimensions
2411 larger than two. It reduces to the ``JordanSpinEJA`` when `B` is
2412 the identity matrix of order ``n``.
2414 We insist that the one-by-one upper-left identity block of `B` be
2415 passed in as well so that we can be passed a matrix of size zero
2416 to construct a trivial algebra.
2420 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
2421 ....: JordanSpinEJA)
2425 When no bilinear form is specified, the identity matrix is used,
2426 and the resulting algebra is the Jordan spin algebra::
2428 sage: B = matrix.identity(AA,3)
2429 sage: J0 = BilinearFormEJA(B)
2430 sage: J1 = JordanSpinEJA(3)
2431 sage: J0.multiplication_table() == J0.multiplication_table()
2434 An error is raised if the matrix `B` does not correspond to a
2435 positive-definite bilinear form::
2437 sage: B = matrix.random(QQ,2,3)
2438 sage: J = BilinearFormEJA(B)
2439 Traceback (most recent call last):
2441 ValueError: bilinear form is not positive-definite
2442 sage: B = matrix.zero(QQ,3)
2443 sage: J = BilinearFormEJA(B)
2444 Traceback (most recent call last):
2446 ValueError: bilinear form is not positive-definite
2450 We can create a zero-dimensional algebra::
2452 sage: B = matrix.identity(AA,0)
2453 sage: J = BilinearFormEJA(B)
2457 We can check the multiplication condition given in the Jordan, von
2458 Neumann, and Wigner paper (and also discussed on my "On the
2459 symmetry..." paper). Note that this relies heavily on the standard
2460 choice of basis, as does anything utilizing the bilinear form
2461 matrix. We opt not to orthonormalize the basis, because if we
2462 did, we would have to normalize the `s_{i}` in a similar manner::
2464 sage: set_random_seed()
2465 sage: n = ZZ.random_element(5)
2466 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
2467 sage: B11 = matrix.identity(QQ,1)
2468 sage: B22 = M.transpose()*M
2469 sage: B = block_matrix(2,2,[ [B11,0 ],
2471 sage: J = BilinearFormEJA(B, orthonormalize=False)
2472 sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
2473 sage: V = J.vector_space()
2474 sage: sis = [ J( V([0] + (M.inverse()*ei).list()).column() )
2475 ....: for ei in eis ]
2476 sage: actual = [ sis[i]*sis[j]
2477 ....: for i in range(n-1)
2478 ....: for j in range(n-1) ]
2479 sage: expected = [ J.one() if i == j else J.zero()
2480 ....: for i in range(n-1)
2481 ....: for j in range(n-1) ]
2482 sage: actual == expected
2486 def __init__(self
, B
, field
=AA
, **kwargs
):
2487 # The matrix "B" is supplied by the user in most cases,
2488 # so it makes sense to check whether or not its positive-
2489 # definite unless we are specifically asked not to...
2490 if ("check_axioms" not in kwargs
) or kwargs
["check_axioms"]:
2491 if not B
.is_positive_definite():
2492 raise ValueError("bilinear form is not positive-definite")
2494 # However, all of the other data for this EJA is computed
2495 # by us in manner that guarantees the axioms are
2496 # satisfied. So, again, unless we are specifically asked to
2497 # verify things, we'll skip the rest of the checks.
2498 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2501 MS
= MatrixSpace(field
, n
, 1)
2503 def inner_product(x
,y
):
2504 return (y
.T
*B
*x
)[0,0]
2506 def jordan_product(x
,y
):
2511 z0
= inner_product(y
,x
)
2512 zbar
= y0
*xbar
+ x0
*ybar
2513 return MS([z0
] + zbar
.list())
2515 column_basis
= tuple( MS(b
) for b
in FreeModule(field
, n
).basis() )
2517 # TODO: I haven't actually checked this, but it seems legit.
2522 super().__init
__(column_basis
,
2527 associative
=associative
,
2530 # The rank of this algebra is two, unless we're in a
2531 # one-dimensional ambient space (because the rank is bounded
2532 # by the ambient dimension).
2533 self
.rank
.set_cache(min(n
,2))
2535 self
.one
.set_cache( self
.zero() )
2537 self
.one
.set_cache( self
.monomial(0) )
2540 def _max_random_instance_size():
2542 The maximum dimension of a random BilinearFormEJA.
2547 def random_instance(cls
, **kwargs
):
2549 Return a random instance of this algebra.
2551 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2553 B
= matrix
.identity(ZZ
, n
)
2554 return cls(B
, **kwargs
)
2556 B11
= matrix
.identity(ZZ
, 1)
2557 M
= matrix
.random(ZZ
, n
-1)
2558 I
= matrix
.identity(ZZ
, n
-1)
2560 while alpha
.is_zero():
2561 alpha
= ZZ
.random_element().abs()
2562 B22
= M
.transpose()*M
+ alpha
*I
2564 from sage
.matrix
.special
import block_matrix
2565 B
= block_matrix(2,2, [ [B11
, ZZ(0) ],
2568 return cls(B
, **kwargs
)
2571 class JordanSpinEJA(BilinearFormEJA
):
2573 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2574 with the usual inner product and jordan product ``x*y =
2575 (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2580 sage: from mjo.eja.eja_algebra import JordanSpinEJA
2584 This multiplication table can be verified by hand::
2586 sage: J = JordanSpinEJA(4)
2587 sage: b0,b1,b2,b3 = J.gens()
2603 We can change the generator prefix::
2605 sage: JordanSpinEJA(2, prefix='B').gens()
2610 Ensure that we have the usual inner product on `R^n`::
2612 sage: set_random_seed()
2613 sage: J = JordanSpinEJA.random_instance()
2614 sage: x,y = J.random_elements(2)
2615 sage: actual = x.inner_product(y)
2616 sage: expected = x.to_vector().inner_product(y.to_vector())
2617 sage: actual == expected
2621 def __init__(self
, n
, *args
, **kwargs
):
2622 # This is a special case of the BilinearFormEJA with the
2623 # identity matrix as its bilinear form.
2624 B
= matrix
.identity(ZZ
, n
)
2626 # Don't orthonormalize because our basis is already
2627 # orthonormal with respect to our inner-product.
2628 if "orthonormalize" not in kwargs
: kwargs
["orthonormalize"] = False
2630 # But also don't pass check_field=False here, because the user
2631 # can pass in a field!
2632 super().__init
__(B
, *args
, **kwargs
)
2635 def _max_random_instance_size():
2637 The maximum dimension of a random JordanSpinEJA.
2642 def random_instance(cls
, **kwargs
):
2644 Return a random instance of this type of algebra.
2646 Needed here to override the implementation for ``BilinearFormEJA``.
2648 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2649 return cls(n
, **kwargs
)
2652 class TrivialEJA(RationalBasisEJA
, ConcreteEJA
):
2654 The trivial Euclidean Jordan algebra consisting of only a zero element.
2658 sage: from mjo.eja.eja_algebra import TrivialEJA
2662 sage: J = TrivialEJA()
2669 sage: 7*J.one()*12*J.one()
2671 sage: J.one().inner_product(J.one())
2673 sage: J.one().norm()
2675 sage: J.one().subalgebra_generated_by()
2676 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
2681 def __init__(self
, field
=AA
, **kwargs
):
2682 jordan_product
= lambda x
,y
: x
2683 inner_product
= lambda x
,y
: field
.zero()
2685 MS
= MatrixSpace(field
,0)
2687 # New defaults for keyword arguments
2688 if "orthonormalize" not in kwargs
: kwargs
["orthonormalize"] = False
2689 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2691 super().__init
__(basis
,
2699 # The rank is zero using my definition, namely the dimension of the
2700 # largest subalgebra generated by any element.
2701 self
.rank
.set_cache(0)
2702 self
.one
.set_cache( self
.zero() )
2705 def random_instance(cls
, **kwargs
):
2706 # We don't take a "size" argument so the superclass method is
2707 # inappropriate for us.
2708 return cls(**kwargs
)
2711 class CartesianProductEJA(FiniteDimensionalEJA
):
2713 The external (orthogonal) direct sum of two or more Euclidean
2714 Jordan algebras. Every Euclidean Jordan algebra decomposes into an
2715 orthogonal direct sum of simple Euclidean Jordan algebras which is
2716 then isometric to a Cartesian product, so no generality is lost by
2717 providing only this construction.
2721 sage: from mjo.eja.eja_algebra import (random_eja,
2722 ....: CartesianProductEJA,
2724 ....: JordanSpinEJA,
2725 ....: RealSymmetricEJA)
2729 The Jordan product is inherited from our factors and implemented by
2730 our CombinatorialFreeModule Cartesian product superclass::
2732 sage: set_random_seed()
2733 sage: J1 = HadamardEJA(2)
2734 sage: J2 = RealSymmetricEJA(2)
2735 sage: J = cartesian_product([J1,J2])
2736 sage: x,y = J.random_elements(2)
2740 The ability to retrieve the original factors is implemented by our
2741 CombinatorialFreeModule Cartesian product superclass::
2743 sage: J1 = HadamardEJA(2, field=QQ)
2744 sage: J2 = JordanSpinEJA(3, field=QQ)
2745 sage: J = cartesian_product([J1,J2])
2746 sage: J.cartesian_factors()
2747 (Euclidean Jordan algebra of dimension 2 over Rational Field,
2748 Euclidean Jordan algebra of dimension 3 over Rational Field)
2750 You can provide more than two factors::
2752 sage: J1 = HadamardEJA(2)
2753 sage: J2 = JordanSpinEJA(3)
2754 sage: J3 = RealSymmetricEJA(3)
2755 sage: cartesian_product([J1,J2,J3])
2756 Euclidean Jordan algebra of dimension 2 over Algebraic Real
2757 Field (+) Euclidean Jordan algebra of dimension 3 over Algebraic
2758 Real Field (+) Euclidean Jordan algebra of dimension 6 over
2759 Algebraic Real Field
2761 Rank is additive on a Cartesian product::
2763 sage: J1 = HadamardEJA(1)
2764 sage: J2 = RealSymmetricEJA(2)
2765 sage: J = cartesian_product([J1,J2])
2766 sage: J1.rank.clear_cache()
2767 sage: J2.rank.clear_cache()
2768 sage: J.rank.clear_cache()
2771 sage: J.rank() == J1.rank() + J2.rank()
2774 The same rank computation works over the rationals, with whatever
2777 sage: J1 = HadamardEJA(1, field=QQ, orthonormalize=False)
2778 sage: J2 = RealSymmetricEJA(2, field=QQ, orthonormalize=False)
2779 sage: J = cartesian_product([J1,J2])
2780 sage: J1.rank.clear_cache()
2781 sage: J2.rank.clear_cache()
2782 sage: J.rank.clear_cache()
2785 sage: J.rank() == J1.rank() + J2.rank()
2788 The product algebra will be associative if and only if all of its
2789 components are associative::
2791 sage: J1 = HadamardEJA(2)
2792 sage: J1.is_associative()
2794 sage: J2 = HadamardEJA(3)
2795 sage: J2.is_associative()
2797 sage: J3 = RealSymmetricEJA(3)
2798 sage: J3.is_associative()
2800 sage: CP1 = cartesian_product([J1,J2])
2801 sage: CP1.is_associative()
2803 sage: CP2 = cartesian_product([J1,J3])
2804 sage: CP2.is_associative()
2807 Cartesian products of Cartesian products work::
2809 sage: J1 = JordanSpinEJA(1)
2810 sage: J2 = JordanSpinEJA(1)
2811 sage: J3 = JordanSpinEJA(1)
2812 sage: J = cartesian_product([J1,cartesian_product([J2,J3])])
2813 sage: J.multiplication_table()
2814 +----++----+----+----+
2815 | * || b0 | b1 | b2 |
2816 +====++====+====+====+
2817 | b0 || b0 | 0 | 0 |
2818 +----++----+----+----+
2819 | b1 || 0 | b1 | 0 |
2820 +----++----+----+----+
2821 | b2 || 0 | 0 | b2 |
2822 +----++----+----+----+
2823 sage: HadamardEJA(3).multiplication_table()
2824 +----++----+----+----+
2825 | * || b0 | b1 | b2 |
2826 +====++====+====+====+
2827 | b0 || b0 | 0 | 0 |
2828 +----++----+----+----+
2829 | b1 || 0 | b1 | 0 |
2830 +----++----+----+----+
2831 | b2 || 0 | 0 | b2 |
2832 +----++----+----+----+
2836 All factors must share the same base field::
2838 sage: J1 = HadamardEJA(2, field=QQ)
2839 sage: J2 = RealSymmetricEJA(2)
2840 sage: CartesianProductEJA((J1,J2))
2841 Traceback (most recent call last):
2843 ValueError: all factors must share the same base field
2845 The cached unit element is the same one that would be computed::
2847 sage: set_random_seed() # long time
2848 sage: J1 = random_eja() # long time
2849 sage: J2 = random_eja() # long time
2850 sage: J = cartesian_product([J1,J2]) # long time
2851 sage: actual = J.one() # long time
2852 sage: J.one.clear_cache() # long time
2853 sage: expected = J.one() # long time
2854 sage: actual == expected # long time
2858 Element
= FiniteDimensionalEJAElement
2861 def __init__(self
, factors
, **kwargs
):
2866 self
._sets
= factors
2868 field
= factors
[0].base_ring()
2869 if not all( J
.base_ring() == field
for J
in factors
):
2870 raise ValueError("all factors must share the same base field")
2872 associative
= all( f
.is_associative() for f
in factors
)
2874 # Compute my matrix space. This category isn't perfect, but
2875 # is good enough for what we need to do.
2876 MS_cat
= MagmaticAlgebras(field
).FiniteDimensional().WithBasis()
2877 MS_cat
= MS_cat
.Unital().CartesianProducts()
2878 MS_factors
= tuple( J
.matrix_space() for J
in factors
)
2879 from sage
.sets
.cartesian_product
import CartesianProduct
2880 MS
= CartesianProduct(MS_factors
, MS_cat
)
2885 for b
in factors
[i
].matrix_basis():
2890 basis
= tuple( MS(b
) for b
in basis
)
2892 # Define jordan/inner products that operate on that matrix_basis.
2893 def jordan_product(x
,y
):
2895 (factors
[i
](x
[i
])*factors
[i
](y
[i
])).to_matrix()
2899 def inner_product(x
, y
):
2901 factors
[i
](x
[i
]).inner_product(factors
[i
](y
[i
]))
2905 # There's no need to check the field since it already came
2906 # from an EJA. Likewise the axioms are guaranteed to be
2907 # satisfied, unless the guy writing this class sucks.
2909 # If you want the basis to be orthonormalized, orthonormalize
2911 FiniteDimensionalEJA
.__init
__(self
,
2917 orthonormalize
=False,
2918 associative
=associative
,
2919 cartesian_product
=True,
2923 self
.rank
.set_cache(sum(J
.rank() for J
in factors
))
2924 ones
= tuple(J
.one().to_matrix() for J
in factors
)
2925 self
.one
.set_cache(self(ones
))
2927 def cartesian_factors(self
):
2928 # Copy/pasted from CombinatorialFreeModule_CartesianProduct.
2931 def cartesian_factor(self
, i
):
2933 Return the ``i``th factor of this algebra.
2935 return self
._sets
[i
]
2938 # Copy/pasted from CombinatorialFreeModule_CartesianProduct.
2939 from sage
.categories
.cartesian_product
import cartesian_product
2940 return cartesian_product
.symbol
.join("%s" % factor
2941 for factor
in self
._sets
)
2943 def matrix_space(self
):
2945 Return the space that our matrix basis lives in as a Cartesian
2948 We don't simply use the ``cartesian_product()`` functor here
2949 because it acts differently on SageMath MatrixSpaces and our
2950 custom MatrixAlgebras, which are CombinatorialFreeModules. We
2951 always want the result to be represented (and indexed) as
2956 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
2958 ....: OctonionHermitianEJA,
2959 ....: RealSymmetricEJA)
2963 sage: J1 = HadamardEJA(1)
2964 sage: J2 = RealSymmetricEJA(2)
2965 sage: J = cartesian_product([J1,J2])
2966 sage: J.matrix_space()
2967 The Cartesian product of (Full MatrixSpace of 1 by 1 dense
2968 matrices over Algebraic Real Field, Full MatrixSpace of 2
2969 by 2 dense matrices over Algebraic Real Field)
2973 sage: J1 = ComplexHermitianEJA(1)
2974 sage: J2 = ComplexHermitianEJA(1)
2975 sage: J = cartesian_product([J1,J2])
2976 sage: J.one().to_matrix()[0]
2980 sage: J.one().to_matrix()[1]
2987 sage: J1 = OctonionHermitianEJA(1)
2988 sage: J2 = OctonionHermitianEJA(1)
2989 sage: J = cartesian_product([J1,J2])
2990 sage: J.one().to_matrix()[0]
2994 sage: J.one().to_matrix()[1]
3000 return super().matrix_space()
3004 def cartesian_projection(self
, i
):
3008 sage: from mjo.eja.eja_algebra import (random_eja,
3009 ....: JordanSpinEJA,
3011 ....: RealSymmetricEJA,
3012 ....: ComplexHermitianEJA)
3016 The projection morphisms are Euclidean Jordan algebra
3019 sage: J1 = HadamardEJA(2)
3020 sage: J2 = RealSymmetricEJA(2)
3021 sage: J = cartesian_product([J1,J2])
3022 sage: J.cartesian_projection(0)
3023 Linear operator between finite-dimensional Euclidean Jordan
3024 algebras represented by the matrix:
3027 Domain: Euclidean Jordan algebra of dimension 2 over Algebraic
3028 Real Field (+) Euclidean Jordan algebra of dimension 3 over
3029 Algebraic Real Field
3030 Codomain: Euclidean Jordan algebra of dimension 2 over Algebraic
3032 sage: J.cartesian_projection(1)
3033 Linear operator between finite-dimensional Euclidean Jordan
3034 algebras represented by the matrix:
3038 Domain: Euclidean Jordan algebra of dimension 2 over Algebraic
3039 Real Field (+) Euclidean Jordan algebra of dimension 3 over
3040 Algebraic Real Field
3041 Codomain: Euclidean Jordan algebra of dimension 3 over Algebraic
3044 The projections work the way you'd expect on the vector
3045 representation of an element::
3047 sage: J1 = JordanSpinEJA(2)
3048 sage: J2 = ComplexHermitianEJA(2)
3049 sage: J = cartesian_product([J1,J2])
3050 sage: pi_left = J.cartesian_projection(0)
3051 sage: pi_right = J.cartesian_projection(1)
3052 sage: pi_left(J.one()).to_vector()
3054 sage: pi_right(J.one()).to_vector()
3056 sage: J.one().to_vector()
3061 The answer never changes::
3063 sage: set_random_seed()
3064 sage: J1 = random_eja()
3065 sage: J2 = random_eja()
3066 sage: J = cartesian_product([J1,J2])
3067 sage: P0 = J.cartesian_projection(0)
3068 sage: P1 = J.cartesian_projection(0)
3073 offset
= sum( self
.cartesian_factor(k
).dimension()
3075 Ji
= self
.cartesian_factor(i
)
3076 Pi
= self
._module
_morphism
(lambda j
: Ji
.monomial(j
- offset
),
3079 return FiniteDimensionalEJAOperator(self
,Ji
,Pi
.matrix())
3082 def cartesian_embedding(self
, i
):
3086 sage: from mjo.eja.eja_algebra import (random_eja,
3087 ....: JordanSpinEJA,
3089 ....: RealSymmetricEJA)
3093 The embedding morphisms are Euclidean Jordan algebra
3096 sage: J1 = HadamardEJA(2)
3097 sage: J2 = RealSymmetricEJA(2)
3098 sage: J = cartesian_product([J1,J2])
3099 sage: J.cartesian_embedding(0)
3100 Linear operator between finite-dimensional Euclidean Jordan
3101 algebras represented by the matrix:
3107 Domain: Euclidean Jordan algebra of dimension 2 over
3108 Algebraic Real Field
3109 Codomain: Euclidean Jordan algebra of dimension 2 over
3110 Algebraic Real Field (+) Euclidean Jordan algebra of
3111 dimension 3 over Algebraic Real Field
3112 sage: J.cartesian_embedding(1)
3113 Linear operator between finite-dimensional Euclidean Jordan
3114 algebras represented by the matrix:
3120 Domain: Euclidean Jordan algebra of dimension 3 over
3121 Algebraic Real Field
3122 Codomain: Euclidean Jordan algebra of dimension 2 over
3123 Algebraic Real Field (+) Euclidean Jordan algebra of
3124 dimension 3 over Algebraic Real Field
3126 The embeddings work the way you'd expect on the vector
3127 representation of an element::
3129 sage: J1 = JordanSpinEJA(3)
3130 sage: J2 = RealSymmetricEJA(2)
3131 sage: J = cartesian_product([J1,J2])
3132 sage: iota_left = J.cartesian_embedding(0)
3133 sage: iota_right = J.cartesian_embedding(1)
3134 sage: iota_left(J1.zero()) == J.zero()
3136 sage: iota_right(J2.zero()) == J.zero()
3138 sage: J1.one().to_vector()
3140 sage: iota_left(J1.one()).to_vector()
3142 sage: J2.one().to_vector()
3144 sage: iota_right(J2.one()).to_vector()
3146 sage: J.one().to_vector()
3151 The answer never changes::
3153 sage: set_random_seed()
3154 sage: J1 = random_eja()
3155 sage: J2 = random_eja()
3156 sage: J = cartesian_product([J1,J2])
3157 sage: E0 = J.cartesian_embedding(0)
3158 sage: E1 = J.cartesian_embedding(0)
3162 Composing a projection with the corresponding inclusion should
3163 produce the identity map, and mismatching them should produce
3166 sage: set_random_seed()
3167 sage: J1 = random_eja()
3168 sage: J2 = random_eja()
3169 sage: J = cartesian_product([J1,J2])
3170 sage: iota_left = J.cartesian_embedding(0)
3171 sage: iota_right = J.cartesian_embedding(1)
3172 sage: pi_left = J.cartesian_projection(0)
3173 sage: pi_right = J.cartesian_projection(1)
3174 sage: pi_left*iota_left == J1.one().operator()
3176 sage: pi_right*iota_right == J2.one().operator()
3178 sage: (pi_left*iota_right).is_zero()
3180 sage: (pi_right*iota_left).is_zero()
3184 offset
= sum( self
.cartesian_factor(k
).dimension()
3186 Ji
= self
.cartesian_factor(i
)
3187 Ei
= Ji
._module
_morphism
(lambda j
: self
.monomial(j
+ offset
),
3189 return FiniteDimensionalEJAOperator(Ji
,self
,Ei
.matrix())
3193 FiniteDimensionalEJA
.CartesianProduct
= CartesianProductEJA
3195 class RationalBasisCartesianProductEJA(CartesianProductEJA
,
3198 A separate class for products of algebras for which we know a
3203 sage: from mjo.eja.eja_algebra import (HadamardEJA,
3204 ....: JordanSpinEJA,
3205 ....: OctonionHermitianEJA,
3206 ....: RealSymmetricEJA)
3210 This gives us fast characteristic polynomial computations in
3211 product algebras, too::
3214 sage: J1 = JordanSpinEJA(2)
3215 sage: J2 = RealSymmetricEJA(3)
3216 sage: J = cartesian_product([J1,J2])
3217 sage: J.characteristic_polynomial_of().degree()
3224 The ``cartesian_product()`` function only uses the first factor to
3225 decide where the result will live; thus we have to be careful to
3226 check that all factors do indeed have a `_rational_algebra` member
3227 before we try to access it::
3229 sage: J1 = OctonionHermitianEJA(1) # no rational basis
3230 sage: J2 = HadamardEJA(2)
3231 sage: cartesian_product([J1,J2])
3232 Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
3233 (+) Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
3234 sage: cartesian_product([J2,J1])
3235 Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
3236 (+) Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
3239 def __init__(self
, algebras
, **kwargs
):
3240 CartesianProductEJA
.__init
__(self
, algebras
, **kwargs
)
3242 self
._rational
_algebra
= None
3243 if self
.vector_space().base_field() is not QQ
:
3244 if all( hasattr(r
, "_rational_algebra") for r
in algebras
):
3245 self
._rational
_algebra
= cartesian_product([
3246 r
._rational
_algebra
for r
in algebras
3250 RationalBasisEJA
.CartesianProduct
= RationalBasisCartesianProductEJA
3252 def random_eja(*args
, **kwargs
):
3253 J1
= ConcreteEJA
.random_instance(*args
, **kwargs
)
3255 # This might make Cartesian products appear roughly as often as
3256 # any other ConcreteEJA.
3257 if ZZ
.random_element(len(ConcreteEJA
.__subclasses
__()) + 1) == 0:
3258 # Use random_eja() again so we can get more than two factors.
3259 J2
= random_eja(*args
, **kwargs
)
3260 J
= cartesian_product([J1
,J2
])