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 FiniteDimensionalEuclideanJordanAlgebraElement
33 from mjo
.eja
.eja_operator
import FiniteDimensionalEuclideanJordanAlgebraOperator
34 from mjo
.eja
.eja_utils
import _mat2vec
36 class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule
):
38 The lowest-level class for representing a Euclidean Jordan algebra.
40 def _coerce_map_from_base_ring(self
):
42 Disable the map from the base ring into the algebra.
44 Performing a nonsense conversion like this automatically
45 is counterpedagogical. The fallback is to try the usual
46 element constructor, which should also fail.
50 sage: from mjo.eja.eja_algebra import random_eja
54 sage: set_random_seed()
55 sage: J = random_eja()
57 Traceback (most recent call last):
59 ValueError: not an element of this algebra
76 * field -- the scalar field for this algebra (must be real)
78 * multiplication_table -- the multiplication table for this
79 algebra's implicit basis. Only the lower-triangular portion
80 of the table is used, since the multiplication is assumed
85 sage: from mjo.eja.eja_algebra import (
86 ....: FiniteDimensionalEuclideanJordanAlgebra,
92 By definition, Jordan multiplication commutes::
94 sage: set_random_seed()
95 sage: J = random_eja()
96 sage: x,y = J.random_elements(2)
100 An error is raised if the Jordan product is not commutative::
102 sage: JP = ((1,2),(0,0))
103 sage: IP = ((1,0),(0,1))
104 sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,JP,IP)
105 Traceback (most recent call last):
107 ValueError: Jordan product is not commutative
109 An error is raised if the inner-product is not commutative::
111 sage: JP = ((1,0),(0,1))
112 sage: IP = ((1,2),(0,0))
113 sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,JP,IP)
114 Traceback (most recent call last):
116 ValueError: inner-product is not commutative
120 The ``field`` we're given must be real with ``check_field=True``::
122 sage: JordanSpinEJA(2, field=QQbar)
123 Traceback (most recent call last):
125 ValueError: scalar field is not real
126 sage: JordanSpinEJA(2, field=QQbar, check_field=False)
127 Euclidean Jordan algebra of dimension 2 over Algebraic Field
129 The multiplication table must be square with ``check_axioms=True``::
131 sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),()),((1,),))
132 Traceback (most recent call last):
134 ValueError: multiplication table is not square
136 The multiplication and inner-product tables must be the same
137 size (and in particular, the inner-product table must also be
138 square) with ``check_axioms=True``::
140 sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((1,),),(()))
141 Traceback (most recent call last):
143 ValueError: multiplication and inner-product tables are
145 sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((1,),),((1,2),))
146 Traceback (most recent call last):
148 ValueError: multiplication and inner-product tables are
153 if not field
.is_subring(RR
):
154 # Note: this does return true for the real algebraic
155 # field, the rationals, and any quadratic field where
156 # we've specified a real embedding.
157 raise ValueError("scalar field is not real")
160 # The multiplication and inner-product tables should be square
161 # if the user wants us to verify them. And we verify them as
162 # soon as possible, because we want to exploit their symmetry.
163 n
= len(multiplication_table
)
165 if not all( len(l
) == n
for l
in multiplication_table
):
166 raise ValueError("multiplication table is not square")
168 # If the multiplication table is square, we can check if
169 # the inner-product table is square by comparing it to the
170 # multiplication table's dimensions.
171 msg
= "multiplication and inner-product tables are different sizes"
172 if not len(inner_product_table
) == n
:
173 raise ValueError(msg
)
175 if not all( len(l
) == n
for l
in inner_product_table
):
176 raise ValueError(msg
)
178 # Check commutativity of the Jordan product (symmetry of
179 # the multiplication table) and the commutativity of the
180 # inner-product (symmetry of the inner-product table)
181 # first if we're going to check them at all.. This has to
182 # be done before we define product_on_basis(), because
183 # that method assumes that self._multiplication_table is
184 # symmetric. And it has to be done before we build
185 # self._inner_product_matrix, because the process used to
186 # construct it assumes symmetry as well.
187 if not all( multiplication_table
[j
][i
]
188 == multiplication_table
[i
][j
]
190 for j
in range(i
+1) ):
191 raise ValueError("Jordan product is not commutative")
193 if not all( inner_product_table
[j
][i
]
194 == inner_product_table
[i
][j
]
196 for j
in range(i
+1) ):
197 raise ValueError("inner-product is not commutative")
199 self
._matrix
_basis
= matrix_basis
202 category
= MagmaticAlgebras(field
).FiniteDimensional()
203 category
= category
.WithBasis().Unital()
205 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
210 self
.print_options(bracket
='')
212 # The multiplication table we're given is necessarily in terms
213 # of vectors, because we don't have an algebra yet for
214 # anything to be an element of. However, it's faster in the
215 # long run to have the multiplication table be in terms of
216 # algebra elements. We do this after calling the superclass
217 # constructor so that from_vector() knows what to do.
219 # Note: we take advantage of symmetry here, and only store
220 # the lower-triangular portion of the table.
221 self
._multiplication
_table
= [ [ self
.vector_space().zero()
222 for j
in range(i
+1) ]
227 elt
= self
.from_vector(multiplication_table
[i
][j
])
228 self
._multiplication
_table
[i
][j
] = elt
230 self
._multiplication
_table
= tuple(map(tuple, self
._multiplication
_table
))
232 # Save our inner product as a matrix, since the efficiency of
233 # matrix multiplication will usually outweigh the fact that we
234 # have to store a redundant upper- or lower-triangular part.
235 # Pre-cache the fact that these are Hermitian (real symmetric,
236 # in fact) in case some e.g. matrix multiplication routine can
237 # take advantage of it.
238 ip_matrix_constructor
= lambda i
,j
: inner_product_table
[i
][j
] if j
<= i
else inner_product_table
[j
][i
]
239 self
._inner
_product
_matrix
= matrix(field
, n
, ip_matrix_constructor
)
240 self
._inner
_product
_matrix
._cache
= {'hermitian': True}
241 self
._inner
_product
_matrix
.set_immutable()
244 if not self
._is
_jordanian
():
245 raise ValueError("Jordan identity does not hold")
246 if not self
._inner
_product
_is
_associative
():
247 raise ValueError("inner product is not associative")
249 def _element_constructor_(self
, elt
):
251 Construct an element of this algebra from its vector or matrix
254 This gets called only after the parent element _call_ method
255 fails to find a coercion for the argument.
259 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
261 ....: RealSymmetricEJA)
265 The identity in `S^n` is converted to the identity in the EJA::
267 sage: J = RealSymmetricEJA(3)
268 sage: I = matrix.identity(QQ,3)
269 sage: J(I) == J.one()
272 This skew-symmetric matrix can't be represented in the EJA::
274 sage: J = RealSymmetricEJA(3)
275 sage: A = matrix(QQ,3, lambda i,j: i-j)
277 Traceback (most recent call last):
279 ValueError: not an element of this algebra
283 Ensure that we can convert any element of the two non-matrix
284 simple algebras (whose matrix representations are columns)
285 back and forth faithfully::
287 sage: set_random_seed()
288 sage: J = HadamardEJA.random_instance()
289 sage: x = J.random_element()
290 sage: J(x.to_vector().column()) == x
292 sage: J = JordanSpinEJA.random_instance()
293 sage: x = J.random_element()
294 sage: J(x.to_vector().column()) == x
298 msg
= "not an element of this algebra"
300 # The superclass implementation of random_element()
301 # needs to be able to coerce "0" into the algebra.
303 elif elt
in self
.base_ring():
304 # Ensure that no base ring -> algebra coercion is performed
305 # by this method. There's some stupidity in sage that would
306 # otherwise propagate to this method; for example, sage thinks
307 # that the integer 3 belongs to the space of 2-by-2 matrices.
308 raise ValueError(msg
)
310 if elt
not in self
.matrix_space():
311 raise ValueError(msg
)
313 # Thanks for nothing! Matrix spaces aren't vector spaces in
314 # Sage, so we have to figure out its matrix-basis coordinates
315 # ourselves. We use the basis space's ring instead of the
316 # element's ring because the basis space might be an algebraic
317 # closure whereas the base ring of the 3-by-3 identity matrix
318 # could be QQ instead of QQbar.
320 # We pass check=False because the matrix basis is "guaranteed"
321 # to be linearly independent... right? Ha ha.
322 V
= VectorSpace(self
.base_ring(), elt
.nrows()*elt
.ncols())
323 W
= V
.span_of_basis( (_mat2vec(s
) for s
in self
.matrix_basis()),
327 coords
= W
.coordinate_vector(_mat2vec(elt
))
328 except ArithmeticError: # vector is not in free module
329 raise ValueError(msg
)
331 return self
.from_vector(coords
)
335 Return a string representation of ``self``.
339 sage: from mjo.eja.eja_algebra import JordanSpinEJA
343 Ensure that it says what we think it says::
345 sage: JordanSpinEJA(2, field=AA)
346 Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
347 sage: JordanSpinEJA(3, field=RDF)
348 Euclidean Jordan algebra of dimension 3 over Real Double Field
351 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
352 return fmt
.format(self
.dimension(), self
.base_ring())
354 def product_on_basis(self
, i
, j
):
355 # We only stored the lower-triangular portion of the
356 # multiplication table.
358 return self
._multiplication
_table
[i
][j
]
360 return self
._multiplication
_table
[j
][i
]
362 def _is_commutative(self
):
364 Whether or not this algebra's multiplication table is commutative.
366 This method should of course always return ``True``, unless
367 this algebra was constructed with ``check_axioms=False`` and
368 passed an invalid multiplication table.
370 return all( self
.product_on_basis(i
,j
) == self
.product_on_basis(i
,j
)
371 for i
in range(self
.dimension())
372 for j
in range(self
.dimension()) )
374 def _is_jordanian(self
):
376 Whether or not this algebra's multiplication table respects the
377 Jordan identity `(x^{2})(xy) = x(x^{2}y)`.
379 We only check one arrangement of `x` and `y`, so for a
380 ``True`` result to be truly true, you should also check
381 :meth:`_is_commutative`. This method should of course always
382 return ``True``, unless this algebra was constructed with
383 ``check_axioms=False`` and passed an invalid multiplication table.
385 return all( (self
.monomial(i
)**2)*(self
.monomial(i
)*self
.monomial(j
))
387 (self
.monomial(i
))*((self
.monomial(i
)**2)*self
.monomial(j
))
388 for i
in range(self
.dimension())
389 for j
in range(self
.dimension()) )
391 def _inner_product_is_associative(self
):
393 Return whether or not this algebra's inner product `B` is
394 associative; that is, whether or not `B(xy,z) = B(x,yz)`.
396 This method should of course always return ``True``, unless
397 this algebra was constructed with ``check_axioms=False`` and
398 passed an invalid multiplication table.
401 # Used to check whether or not something is zero in an inexact
402 # ring. This number is sufficient to allow the construction of
403 # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True.
406 for i
in range(self
.dimension()):
407 for j
in range(self
.dimension()):
408 for k
in range(self
.dimension()):
412 diff
= (x
*y
).inner_product(z
) - x
.inner_product(y
*z
)
414 if self
.base_ring().is_exact():
418 if diff
.abs() > epsilon
:
424 def characteristic_polynomial_of(self
):
426 Return the algebra's "characteristic polynomial of" function,
427 which is itself a multivariate polynomial that, when evaluated
428 at the coordinates of some algebra element, returns that
429 element's characteristic polynomial.
431 The resulting polynomial has `n+1` variables, where `n` is the
432 dimension of this algebra. The first `n` variables correspond to
433 the coordinates of an algebra element: when evaluated at the
434 coordinates of an algebra element with respect to a certain
435 basis, the result is a univariate polynomial (in the one
436 remaining variable ``t``), namely the characteristic polynomial
441 sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA
445 The characteristic polynomial in the spin algebra is given in
446 Alizadeh, Example 11.11::
448 sage: J = JordanSpinEJA(3)
449 sage: p = J.characteristic_polynomial_of(); p
450 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
451 sage: xvec = J.one().to_vector()
455 By definition, the characteristic polynomial is a monic
456 degree-zero polynomial in a rank-zero algebra. Note that
457 Cayley-Hamilton is indeed satisfied since the polynomial
458 ``1`` evaluates to the identity element of the algebra on
461 sage: J = TrivialEJA()
462 sage: J.characteristic_polynomial_of()
469 # The list of coefficient polynomials a_0, a_1, a_2, ..., a_(r-1).
470 a
= self
._charpoly
_coefficients
()
472 # We go to a bit of trouble here to reorder the
473 # indeterminates, so that it's easier to evaluate the
474 # characteristic polynomial at x's coordinates and get back
475 # something in terms of t, which is what we want.
476 S
= PolynomialRing(self
.base_ring(),'t')
480 S
= PolynomialRing(S
, R
.variable_names())
483 return (t
**r
+ sum( a
[k
]*(t
**k
) for k
in range(r
) ))
485 def coordinate_polynomial_ring(self
):
487 The multivariate polynomial ring in which this algebra's
488 :meth:`characteristic_polynomial_of` lives.
492 sage: from mjo.eja.eja_algebra import (HadamardEJA,
493 ....: RealSymmetricEJA)
497 sage: J = HadamardEJA(2)
498 sage: J.coordinate_polynomial_ring()
499 Multivariate Polynomial Ring in X1, X2...
500 sage: J = RealSymmetricEJA(3,field=QQ,orthonormalize=False)
501 sage: J.coordinate_polynomial_ring()
502 Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6...
505 var_names
= tuple( "X%d" % z
for z
in range(1, self
.dimension()+1) )
506 return PolynomialRing(self
.base_ring(), var_names
)
508 def inner_product(self
, x
, y
):
510 The inner product associated with this Euclidean Jordan algebra.
512 Defaults to the trace inner product, but can be overridden by
513 subclasses if they are sure that the necessary properties are
518 sage: from mjo.eja.eja_algebra import (random_eja,
520 ....: BilinearFormEJA)
524 Our inner product is "associative," which means the following for
525 a symmetric bilinear form::
527 sage: set_random_seed()
528 sage: J = random_eja()
529 sage: x,y,z = J.random_elements(3)
530 sage: (x*y).inner_product(z) == y.inner_product(x*z)
535 Ensure that this is the usual inner product for the algebras
538 sage: set_random_seed()
539 sage: J = HadamardEJA.random_instance()
540 sage: x,y = J.random_elements(2)
541 sage: actual = x.inner_product(y)
542 sage: expected = x.to_vector().inner_product(y.to_vector())
543 sage: actual == expected
546 Ensure that this is one-half of the trace inner-product in a
547 BilinearFormEJA that isn't just the reals (when ``n`` isn't
548 one). This is in Faraut and Koranyi, and also my "On the
551 sage: set_random_seed()
552 sage: J = BilinearFormEJA.random_instance()
553 sage: n = J.dimension()
554 sage: x = J.random_element()
555 sage: y = J.random_element()
556 sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2)
559 B
= self
._inner
_product
_matrix
560 return (B
*x
.to_vector()).inner_product(y
.to_vector())
563 def is_trivial(self
):
565 Return whether or not this algebra is trivial.
567 A trivial algebra contains only the zero element.
571 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
576 sage: J = ComplexHermitianEJA(3)
582 sage: J = TrivialEJA()
587 return self
.dimension() == 0
590 def multiplication_table(self
):
592 Return a visual representation of this algebra's multiplication
593 table (on basis elements).
597 sage: from mjo.eja.eja_algebra import JordanSpinEJA
601 sage: J = JordanSpinEJA(4)
602 sage: J.multiplication_table()
603 +----++----+----+----+----+
604 | * || e0 | e1 | e2 | e3 |
605 +====++====+====+====+====+
606 | e0 || e0 | e1 | e2 | e3 |
607 +----++----+----+----+----+
608 | e1 || e1 | e0 | 0 | 0 |
609 +----++----+----+----+----+
610 | e2 || e2 | 0 | e0 | 0 |
611 +----++----+----+----+----+
612 | e3 || e3 | 0 | 0 | e0 |
613 +----++----+----+----+----+
617 M
= [ [ self
.zero() for j
in range(n
) ]
621 M
[i
][j
] = self
._multiplication
_table
[i
][j
]
625 # Prepend the left "header" column entry Can't do this in
626 # the loop because it messes up the symmetry.
627 M
[i
] = [self
.monomial(i
)] + M
[i
]
629 # Prepend the header row.
630 M
= [["*"] + list(self
.gens())] + M
631 return table(M
, header_row
=True, header_column
=True, frame
=True)
634 def matrix_basis(self
):
636 Return an (often more natural) representation of this algebras
637 basis as an ordered tuple of matrices.
639 Every finite-dimensional Euclidean Jordan Algebra is a, up to
640 Jordan isomorphism, a direct sum of five simple
641 algebras---four of which comprise Hermitian matrices. And the
642 last type of algebra can of course be thought of as `n`-by-`1`
643 column matrices (ambiguusly called column vectors) to avoid
644 special cases. As a result, matrices (and column vectors) are
645 a natural representation format for Euclidean Jordan algebra
648 But, when we construct an algebra from a basis of matrices,
649 those matrix representations are lost in favor of coordinate
650 vectors *with respect to* that basis. We could eventually
651 convert back if we tried hard enough, but having the original
652 representations handy is valuable enough that we simply store
653 them and return them from this method.
655 Why implement this for non-matrix algebras? Avoiding special
656 cases for the :class:`BilinearFormEJA` pays with simplicity in
657 its own right. But mainly, we would like to be able to assume
658 that elements of a :class:`DirectSumEJA` can be displayed
659 nicely, without having to have special classes for direct sums
660 one of whose components was a matrix algebra.
664 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
665 ....: RealSymmetricEJA)
669 sage: J = RealSymmetricEJA(2)
671 Finite family {0: e0, 1: e1, 2: e2}
672 sage: J.matrix_basis()
674 [1 0] [ 0 0.7071067811865475?] [0 0]
675 [0 0], [0.7071067811865475? 0], [0 1]
680 sage: J = JordanSpinEJA(2)
682 Finite family {0: e0, 1: e1}
683 sage: J.matrix_basis()
689 if self
._matrix
_basis
is None:
690 M
= self
.matrix_space()
691 return tuple( M(b
.to_vector()) for b
in self
.basis() )
693 return self
._matrix
_basis
696 def matrix_space(self
):
698 Return the matrix space in which this algebra's elements live, if
699 we think of them as matrices (including column vectors of the
702 Generally this will be an `n`-by-`1` column-vector space,
703 except when the algebra is trivial. There it's `n`-by-`n`
704 (where `n` is zero), to ensure that two elements of the matrix
705 space (empty matrices) can be multiplied.
707 Matrix algebras override this with something more useful.
709 if self
.is_trivial():
710 return MatrixSpace(self
.base_ring(), 0)
711 elif self
._matrix
_basis
is None or len(self
._matrix
_basis
) == 0:
712 return MatrixSpace(self
.base_ring(), self
.dimension(), 1)
714 return self
._matrix
_basis
[0].matrix_space()
720 Return the unit element of this algebra.
724 sage: from mjo.eja.eja_algebra import (HadamardEJA,
729 sage: J = HadamardEJA(5)
731 e0 + e1 + e2 + e3 + e4
735 The identity element acts like the identity::
737 sage: set_random_seed()
738 sage: J = random_eja()
739 sage: x = J.random_element()
740 sage: J.one()*x == x and x*J.one() == x
743 The matrix of the unit element's operator is the identity::
745 sage: set_random_seed()
746 sage: J = random_eja()
747 sage: actual = J.one().operator().matrix()
748 sage: expected = matrix.identity(J.base_ring(), J.dimension())
749 sage: actual == expected
752 Ensure that the cached unit element (often precomputed by
753 hand) agrees with the computed one::
755 sage: set_random_seed()
756 sage: J = random_eja()
757 sage: cached = J.one()
758 sage: J.one.clear_cache()
759 sage: J.one() == cached
763 # We can brute-force compute the matrices of the operators
764 # that correspond to the basis elements of this algebra.
765 # If some linear combination of those basis elements is the
766 # algebra identity, then the same linear combination of
767 # their matrices has to be the identity matrix.
769 # Of course, matrices aren't vectors in sage, so we have to
770 # appeal to the "long vectors" isometry.
771 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
773 # Now we use basic linear algebra to find the coefficients,
774 # of the matrices-as-vectors-linear-combination, which should
775 # work for the original algebra basis too.
776 A
= matrix(self
.base_ring(), oper_vecs
)
778 # We used the isometry on the left-hand side already, but we
779 # still need to do it for the right-hand side. Recall that we
780 # wanted something that summed to the identity matrix.
781 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
783 # Now if there's an identity element in the algebra, this
784 # should work. We solve on the left to avoid having to
785 # transpose the matrix "A".
786 return self
.from_vector(A
.solve_left(b
))
789 def peirce_decomposition(self
, c
):
791 The Peirce decomposition of this algebra relative to the
794 In the future, this can be extended to a complete system of
795 orthogonal idempotents.
799 - ``c`` -- an idempotent of this algebra.
803 A triple (J0, J5, J1) containing two subalgebras and one subspace
806 - ``J0`` -- the algebra on the eigenspace of ``c.operator()``
807 corresponding to the eigenvalue zero.
809 - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()``
810 corresponding to the eigenvalue one-half.
812 - ``J1`` -- the algebra on the eigenspace of ``c.operator()``
813 corresponding to the eigenvalue one.
815 These are the only possible eigenspaces for that operator, and this
816 algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are
817 orthogonal, and are subalgebras of this algebra with the appropriate
822 sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA
826 The canonical example comes from the symmetric matrices, which
827 decompose into diagonal and off-diagonal parts::
829 sage: J = RealSymmetricEJA(3)
830 sage: C = matrix(QQ, [ [1,0,0],
834 sage: J0,J5,J1 = J.peirce_decomposition(c)
836 Euclidean Jordan algebra of dimension 1...
838 Vector space of degree 6 and dimension 2...
840 Euclidean Jordan algebra of dimension 3...
841 sage: J0.one().to_matrix()
845 sage: orig_df = AA.options.display_format
846 sage: AA.options.display_format = 'radical'
847 sage: J.from_vector(J5.basis()[0]).to_matrix()
851 sage: J.from_vector(J5.basis()[1]).to_matrix()
855 sage: AA.options.display_format = orig_df
856 sage: J1.one().to_matrix()
863 Every algebra decomposes trivially with respect to its identity
866 sage: set_random_seed()
867 sage: J = random_eja()
868 sage: J0,J5,J1 = J.peirce_decomposition(J.one())
869 sage: J0.dimension() == 0 and J5.dimension() == 0
871 sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
874 The decomposition is into eigenspaces, and its components are
875 therefore necessarily orthogonal. Moreover, the identity
876 elements in the two subalgebras are the projections onto their
877 respective subspaces of the superalgebra's identity element::
879 sage: set_random_seed()
880 sage: J = random_eja()
881 sage: x = J.random_element()
882 sage: if not J.is_trivial():
883 ....: while x.is_nilpotent():
884 ....: x = J.random_element()
885 sage: c = x.subalgebra_idempotent()
886 sage: J0,J5,J1 = J.peirce_decomposition(c)
888 sage: for (w,y,z) in zip(J0.basis(), J5.basis(), J1.basis()):
889 ....: w = w.superalgebra_element()
890 ....: y = J.from_vector(y)
891 ....: z = z.superalgebra_element()
892 ....: ipsum += w.inner_product(y).abs()
893 ....: ipsum += w.inner_product(z).abs()
894 ....: ipsum += y.inner_product(z).abs()
897 sage: J1(c) == J1.one()
899 sage: J0(J.one() - c) == J0.one()
903 if not c
.is_idempotent():
904 raise ValueError("element is not idempotent: %s" % c
)
906 from mjo
.eja
.eja_subalgebra
import FiniteDimensionalEuclideanJordanSubalgebra
908 # Default these to what they should be if they turn out to be
909 # trivial, because eigenspaces_left() won't return eigenvalues
910 # corresponding to trivial spaces (e.g. it returns only the
911 # eigenspace corresponding to lambda=1 if you take the
912 # decomposition relative to the identity element).
913 trivial
= FiniteDimensionalEuclideanJordanSubalgebra(self
, ())
914 J0
= trivial
# eigenvalue zero
915 J5
= VectorSpace(self
.base_ring(), 0) # eigenvalue one-half
916 J1
= trivial
# eigenvalue one
918 for (eigval
, eigspace
) in c
.operator().matrix().right_eigenspaces():
919 if eigval
== ~
(self
.base_ring()(2)):
922 gens
= tuple( self
.from_vector(b
) for b
in eigspace
.basis() )
923 subalg
= FiniteDimensionalEuclideanJordanSubalgebra(self
,
931 raise ValueError("unexpected eigenvalue: %s" % eigval
)
936 def random_element(self
, thorough
=False):
938 Return a random element of this algebra.
940 Our algebra superclass method only returns a linear
941 combination of at most two basis elements. We instead
942 want the vector space "random element" method that
943 returns a more diverse selection.
947 - ``thorough`` -- (boolean; default False) whether or not we
948 should generate irrational coefficients for the random
949 element when our base ring is irrational; this slows the
950 algebra operations to a crawl, but any truly random method
954 # For a general base ring... maybe we can trust this to do the
955 # right thing? Unlikely, but.
956 V
= self
.vector_space()
957 v
= V
.random_element()
959 if self
.base_ring() is AA
:
960 # The "random element" method of the algebraic reals is
961 # stupid at the moment, and only returns integers between
962 # -2 and 2, inclusive:
964 # https://trac.sagemath.org/ticket/30875
966 # Instead, we implement our own "random vector" method,
967 # and then coerce that into the algebra. We use the vector
968 # space degree here instead of the dimension because a
969 # subalgebra could (for example) be spanned by only two
970 # vectors, each with five coordinates. We need to
971 # generate all five coordinates.
973 v
*= QQbar
.random_element().real()
975 v
*= QQ
.random_element()
977 return self
.from_vector(V
.coordinate_vector(v
))
979 def random_elements(self
, count
, thorough
=False):
981 Return ``count`` random elements as a tuple.
985 - ``thorough`` -- (boolean; default False) whether or not we
986 should generate irrational coefficients for the random
987 elements when our base ring is irrational; this slows the
988 algebra operations to a crawl, but any truly random method
993 sage: from mjo.eja.eja_algebra import JordanSpinEJA
997 sage: J = JordanSpinEJA(3)
998 sage: x,y,z = J.random_elements(3)
999 sage: all( [ x in J, y in J, z in J ])
1001 sage: len( J.random_elements(10) ) == 10
1005 return tuple( self
.random_element(thorough
)
1006 for idx
in range(count
) )
1010 def _charpoly_coefficients(self
):
1012 The `r` polynomial coefficients of the "characteristic polynomial
1015 n
= self
.dimension()
1016 R
= self
.coordinate_polynomial_ring()
1018 F
= R
.fraction_field()
1021 # From a result in my book, these are the entries of the
1022 # basis representation of L_x.
1023 return sum( vars[k
]*self
.monomial(k
).operator().matrix()[i
,j
]
1026 L_x
= matrix(F
, n
, n
, L_x_i_j
)
1029 if self
.rank
.is_in_cache():
1031 # There's no need to pad the system with redundant
1032 # columns if we *know* they'll be redundant.
1035 # Compute an extra power in case the rank is equal to
1036 # the dimension (otherwise, we would stop at x^(r-1)).
1037 x_powers
= [ (L_x
**k
)*self
.one().to_vector()
1038 for k
in range(n
+1) ]
1039 A
= matrix
.column(F
, x_powers
[:n
])
1040 AE
= A
.extended_echelon_form()
1047 # The theory says that only the first "r" coefficients are
1048 # nonzero, and they actually live in the original polynomial
1049 # ring and not the fraction field. We negate them because
1050 # in the actual characteristic polynomial, they get moved
1051 # to the other side where x^r lives.
1052 return -A_rref
.solve_right(E
*b
).change_ring(R
)[:r
]
1057 Return the rank of this EJA.
1059 This is a cached method because we know the rank a priori for
1060 all of the algebras we can construct. Thus we can avoid the
1061 expensive ``_charpoly_coefficients()`` call unless we truly
1062 need to compute the whole characteristic polynomial.
1066 sage: from mjo.eja.eja_algebra import (HadamardEJA,
1067 ....: JordanSpinEJA,
1068 ....: RealSymmetricEJA,
1069 ....: ComplexHermitianEJA,
1070 ....: QuaternionHermitianEJA,
1075 The rank of the Jordan spin algebra is always two::
1077 sage: JordanSpinEJA(2).rank()
1079 sage: JordanSpinEJA(3).rank()
1081 sage: JordanSpinEJA(4).rank()
1084 The rank of the `n`-by-`n` Hermitian real, complex, or
1085 quaternion matrices is `n`::
1087 sage: RealSymmetricEJA(4).rank()
1089 sage: ComplexHermitianEJA(3).rank()
1091 sage: QuaternionHermitianEJA(2).rank()
1096 Ensure that every EJA that we know how to construct has a
1097 positive integer rank, unless the algebra is trivial in
1098 which case its rank will be zero::
1100 sage: set_random_seed()
1101 sage: J = random_eja()
1105 sage: r > 0 or (r == 0 and J.is_trivial())
1108 Ensure that computing the rank actually works, since the ranks
1109 of all simple algebras are known and will be cached by default::
1111 sage: set_random_seed() # long time
1112 sage: J = random_eja() # long time
1113 sage: caches = J.rank() # long time
1114 sage: J.rank.clear_cache() # long time
1115 sage: J.rank() == cached # long time
1119 return len(self
._charpoly
_coefficients
())
1122 def vector_space(self
):
1124 Return the vector space that underlies this algebra.
1128 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1132 sage: J = RealSymmetricEJA(2)
1133 sage: J.vector_space()
1134 Vector space of dimension 3 over...
1137 return self
.zero().to_vector().parent().ambient_vector_space()
1140 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
1142 class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
1144 New class for algebras whose supplied basis elements have all rational entries.
1148 sage: from mjo.eja.eja_algebra import BilinearFormEJA
1152 The supplied basis is orthonormalized by default::
1154 sage: B = matrix(QQ, [[1, 0, 0], [0, 25, -32], [0, -32, 41]])
1155 sage: J = BilinearFormEJA(B)
1156 sage: J.matrix_basis()
1169 orthonormalize
=True,
1176 # Abuse the check_field parameter to check that the entries of
1177 # out basis (in ambient coordinates) are in the field QQ.
1178 if not all( all(b_i
in QQ
for b_i
in b
.list()) for b
in basis
):
1179 raise TypeError("basis not rational")
1181 # Temporary(?) hack to ensure that the matrix and vector bases
1182 # are over the same ring.
1183 basis
= tuple( b
.change_ring(field
) for b
in basis
)
1186 vector_basis
= basis
1188 from sage
.structure
.element
import is_Matrix
1189 basis_is_matrices
= False
1193 if is_Matrix(basis
[0]):
1194 basis_is_matrices
= True
1195 from mjo
.eja
.eja_utils
import _vec2mat
1196 vector_basis
= tuple( map(_mat2vec
,basis
) )
1197 degree
= basis
[0].nrows()**2
1199 degree
= basis
[0].degree()
1201 V
= VectorSpace(field
, degree
)
1203 # Save a copy of an algebra with the original, rational basis
1204 # and over QQ where computations are fast.
1205 self
._rational
_algebra
= None
1208 # There's no point in constructing the extra algebra if this
1209 # one is already rational.
1211 # Note: the same Jordan and inner-products work here,
1212 # because they are necessarily defined with respect to
1213 # ambient coordinates and not any particular basis.
1214 self
._rational
_algebra
= RationalBasisEuclideanJordanAlgebra(
1219 orthonormalize
=False,
1226 # Compute the deorthonormalized tables before we orthonormalize
1227 # the given basis. The "check" parameter here guarantees that
1228 # the basis is linearly-independent.
1229 W
= V
.span_of_basis( vector_basis
, check
=check_axioms
)
1231 # Note: the Jordan and inner-products are defined in terms
1232 # of the ambient basis. It's important that their arguments
1233 # are in ambient coordinates as well.
1235 for j
in range(i
+1):
1236 # given basis w.r.t. ambient coords
1237 q_i
= vector_basis
[i
]
1238 q_j
= vector_basis
[j
]
1240 if basis_is_matrices
:
1244 elt
= jordan_product(q_i
, q_j
)
1245 ip
= inner_product(q_i
, q_j
)
1247 if basis_is_matrices
:
1248 # do another mat2vec because the multiplication
1249 # table is in terms of vectors
1252 # We overwrite the name "vector_basis" in a second, but never modify it
1253 # in place, to this effectively makes a copy of it.
1254 deortho_vector_basis
= vector_basis
1255 self
._deortho
_matrix
= None
1258 from mjo
.eja
.eja_utils
import gram_schmidt
1259 if basis_is_matrices
:
1260 vector_ip
= lambda x
,y
: inner_product(_vec2mat(x
), _vec2mat(y
))
1261 vector_basis
= gram_schmidt(vector_basis
, vector_ip
)
1263 vector_basis
= gram_schmidt(vector_basis
, inner_product
)
1265 # Normalize the "matrix" basis, too!
1266 basis
= vector_basis
1268 if basis_is_matrices
:
1269 basis
= tuple( map(_vec2mat
,basis
) )
1271 W
= V
.span_of_basis( vector_basis
, check
=check_axioms
)
1273 # Now "W" is the vector space of our algebra coordinates. The
1274 # variables "X1", "X2",... refer to the entries of vectors in
1275 # W. Thus to convert back and forth between the orthonormal
1276 # coordinates and the given ones, we need to stick the original
1278 U
= V
.span_of_basis( deortho_vector_basis
, check
=check_axioms
)
1279 self
._deortho
_matrix
= matrix( U
.coordinate_vector(q
)
1280 for q
in vector_basis
)
1282 # If the superclass constructor is going to verify the
1283 # symmetry of this table, it has better at least be
1286 mult_table
= [ [0 for j
in range(n
)] for i
in range(n
) ]
1287 ip_table
= [ [0 for j
in range(n
)] for i
in range(n
) ]
1289 mult_table
= [ [0 for j
in range(i
+1)] for i
in range(n
) ]
1290 ip_table
= [ [0 for j
in range(i
+1)] for i
in range(n
) ]
1292 # Note: the Jordan and inner-products are defined in terms
1293 # of the ambient basis. It's important that their arguments
1294 # are in ambient coordinates as well.
1296 for j
in range(i
+1):
1297 # ortho basis w.r.t. ambient coords
1298 q_i
= vector_basis
[i
]
1299 q_j
= vector_basis
[j
]
1301 if basis_is_matrices
:
1305 elt
= jordan_product(q_i
, q_j
)
1306 ip
= inner_product(q_i
, q_j
)
1308 if basis_is_matrices
:
1309 # do another mat2vec because the multiplication
1310 # table is in terms of vectors
1313 elt
= W
.coordinate_vector(elt
)
1314 mult_table
[i
][j
] = elt
1317 # The tables are square if we're verifying that they
1319 mult_table
[j
][i
] = elt
1322 if basis_is_matrices
:
1326 basis
= tuple( x
.column() for x
in basis
)
1328 super().__init
__(field
,
1333 basis
, # matrix basis
1338 def _charpoly_coefficients(self
):
1342 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
1343 ....: JordanSpinEJA)
1347 The base ring of the resulting polynomial coefficients is what
1348 it should be, and not the rationals (unless the algebra was
1349 already over the rationals)::
1351 sage: J = JordanSpinEJA(3)
1352 sage: J._charpoly_coefficients()
1353 (X1^2 - X2^2 - X3^2, -2*X1)
1354 sage: a0 = J._charpoly_coefficients()[0]
1356 Algebraic Real Field
1357 sage: a0.base_ring()
1358 Algebraic Real Field
1361 if self
._rational
_algebra
is None:
1362 # There's no need to construct *another* algebra over the
1363 # rationals if this one is already over the
1364 # rationals. Likewise, if we never orthonormalized our
1365 # basis, we might as well just use the given one.
1366 superclass
= super(RationalBasisEuclideanJordanAlgebra
, self
)
1367 return superclass
._charpoly
_coefficients
()
1369 # Do the computation over the rationals. The answer will be
1370 # the same, because all we've done is a change of basis.
1371 # Then, change back from QQ to our real base ring
1372 a
= ( a_i
.change_ring(self
.base_ring())
1373 for a_i
in self
._rational
_algebra
._charpoly
_coefficients
() )
1375 # Now convert the coordinate variables back to the
1376 # deorthonormalized ones.
1377 R
= self
.coordinate_polynomial_ring()
1378 from sage
.modules
.free_module_element
import vector
1379 X
= vector(R
, R
.gens())
1380 BX
= self
._deortho
_matrix
*X
1382 subs_dict
= { X[i]: BX[i] for i in range(len(X)) }
1383 return tuple( a_i
.subs(subs_dict
) for a_i
in a
)
1385 class ConcreteEuclideanJordanAlgebra(RationalBasisEuclideanJordanAlgebra
):
1387 A class for the Euclidean Jordan algebras that we know by name.
1389 These are the Jordan algebras whose basis, multiplication table,
1390 rank, and so on are known a priori. More to the point, they are
1391 the Euclidean Jordan algebras for which we are able to conjure up
1392 a "random instance."
1396 sage: from mjo.eja.eja_algebra import ConcreteEuclideanJordanAlgebra
1400 Our basis is normalized with respect to the algebra's inner
1401 product, unless we specify otherwise::
1403 sage: set_random_seed()
1404 sage: J = ConcreteEuclideanJordanAlgebra.random_instance()
1405 sage: all( b.norm() == 1 for b in J.gens() )
1408 Since our basis is orthonormal with respect to the algebra's inner
1409 product, and since we know that this algebra is an EJA, any
1410 left-multiplication operator's matrix will be symmetric because
1411 natural->EJA basis representation is an isometry and within the
1412 EJA the operator is self-adjoint by the Jordan axiom::
1414 sage: set_random_seed()
1415 sage: J = ConcreteEuclideanJordanAlgebra.random_instance()
1416 sage: x = J.random_element()
1417 sage: x.operator().is_self_adjoint()
1422 def _max_random_instance_size():
1424 Return an integer "size" that is an upper bound on the size of
1425 this algebra when it is used in a random test
1426 case. Unfortunately, the term "size" is ambiguous -- when
1427 dealing with `R^n` under either the Hadamard or Jordan spin
1428 product, the "size" refers to the dimension `n`. When dealing
1429 with a matrix algebra (real symmetric or complex/quaternion
1430 Hermitian), it refers to the size of the matrix, which is far
1431 less than the dimension of the underlying vector space.
1433 This method must be implemented in each subclass.
1435 raise NotImplementedError
1438 def random_instance(cls
, *args
, **kwargs
):
1440 Return a random instance of this type of algebra.
1442 This method should be implemented in each subclass.
1444 from sage
.misc
.prandom
import choice
1445 eja_class
= choice(cls
.__subclasses
__())
1447 # These all bubble up to the RationalBasisEuclideanJordanAlgebra
1448 # superclass constructor, so any (kw)args valid there are also
1450 return eja_class
.random_instance(*args
, **kwargs
)
1453 class MatrixEuclideanJordanAlgebra
:
1455 def dimension_over_reals():
1457 The dimension of this matrix's base ring over the reals.
1459 The reals are dimension one over themselves, obviously; that's
1460 just `\mathbb{R}^{1}`. Likewise, the complex numbers `a + bi`
1461 have dimension two. Finally, the quaternions have dimension
1462 four over the reals.
1464 This is used to determine the size of the matrix returned from
1465 :meth:`real_embed`, among other things.
1467 raise NotImplementedError
1470 def real_embed(cls
,M
):
1472 Embed the matrix ``M`` into a space of real matrices.
1474 The matrix ``M`` can have entries in any field at the moment:
1475 the real numbers, complex numbers, or quaternions. And although
1476 they are not a field, we can probably support octonions at some
1477 point, too. This function returns a real matrix that "acts like"
1478 the original with respect to matrix multiplication; i.e.
1480 real_embed(M*N) = real_embed(M)*real_embed(N)
1483 if M
.ncols() != M
.nrows():
1484 raise ValueError("the matrix 'M' must be square")
1489 def real_unembed(cls
,M
):
1491 The inverse of :meth:`real_embed`.
1493 if M
.ncols() != M
.nrows():
1494 raise ValueError("the matrix 'M' must be square")
1495 if not ZZ(M
.nrows()).mod(cls
.dimension_over_reals()).is_zero():
1496 raise ValueError("the matrix 'M' must be a real embedding")
1500 def jordan_product(X
,Y
):
1501 return (X
*Y
+ Y
*X
)/2
1504 def trace_inner_product(cls
,X
,Y
):
1506 Compute the trace inner-product of two real-embeddings.
1510 sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
1511 ....: ComplexHermitianEJA,
1512 ....: QuaternionHermitianEJA)
1516 This gives the same answer as it would if we computed the trace
1517 from the unembedded (original) matrices::
1519 sage: set_random_seed()
1520 sage: J = RealSymmetricEJA.random_instance()
1521 sage: x,y = J.random_elements(2)
1522 sage: Xe = x.to_matrix()
1523 sage: Ye = y.to_matrix()
1524 sage: X = J.real_unembed(Xe)
1525 sage: Y = J.real_unembed(Ye)
1526 sage: expected = (X*Y).trace()
1527 sage: actual = J.trace_inner_product(Xe,Ye)
1528 sage: actual == expected
1533 sage: set_random_seed()
1534 sage: J = ComplexHermitianEJA.random_instance()
1535 sage: x,y = J.random_elements(2)
1536 sage: Xe = x.to_matrix()
1537 sage: Ye = y.to_matrix()
1538 sage: X = J.real_unembed(Xe)
1539 sage: Y = J.real_unembed(Ye)
1540 sage: expected = (X*Y).trace().real()
1541 sage: actual = J.trace_inner_product(Xe,Ye)
1542 sage: actual == expected
1547 sage: set_random_seed()
1548 sage: J = QuaternionHermitianEJA.random_instance()
1549 sage: x,y = J.random_elements(2)
1550 sage: Xe = x.to_matrix()
1551 sage: Ye = y.to_matrix()
1552 sage: X = J.real_unembed(Xe)
1553 sage: Y = J.real_unembed(Ye)
1554 sage: expected = (X*Y).trace().coefficient_tuple()[0]
1555 sage: actual = J.trace_inner_product(Xe,Ye)
1556 sage: actual == expected
1560 Xu
= cls
.real_unembed(X
)
1561 Yu
= cls
.real_unembed(Y
)
1562 tr
= (Xu
*Yu
).trace()
1565 # Works in QQ, AA, RDF, et cetera.
1567 except AttributeError:
1568 # A quaternion doesn't have a real() method, but does
1569 # have coefficient_tuple() method that returns the
1570 # coefficients of 1, i, j, and k -- in that order.
1571 return tr
.coefficient_tuple()[0]
1574 class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1576 def dimension_over_reals():
1580 class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra
,
1581 RealMatrixEuclideanJordanAlgebra
):
1583 The rank-n simple EJA consisting of real symmetric n-by-n
1584 matrices, the usual symmetric Jordan product, and the trace inner
1585 product. It has dimension `(n^2 + n)/2` over the reals.
1589 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1593 sage: J = RealSymmetricEJA(2)
1594 sage: e0, e1, e2 = J.gens()
1602 In theory, our "field" can be any subfield of the reals::
1604 sage: RealSymmetricEJA(2, field=RDF)
1605 Euclidean Jordan algebra of dimension 3 over Real Double Field
1606 sage: RealSymmetricEJA(2, field=RR)
1607 Euclidean Jordan algebra of dimension 3 over Real Field with
1608 53 bits of precision
1612 The dimension of this algebra is `(n^2 + n) / 2`::
1614 sage: set_random_seed()
1615 sage: n_max = RealSymmetricEJA._max_random_instance_size()
1616 sage: n = ZZ.random_element(1, n_max)
1617 sage: J = RealSymmetricEJA(n)
1618 sage: J.dimension() == (n^2 + n)/2
1621 The Jordan multiplication is what we think it is::
1623 sage: set_random_seed()
1624 sage: J = RealSymmetricEJA.random_instance()
1625 sage: x,y = J.random_elements(2)
1626 sage: actual = (x*y).to_matrix()
1627 sage: X = x.to_matrix()
1628 sage: Y = y.to_matrix()
1629 sage: expected = (X*Y + Y*X)/2
1630 sage: actual == expected
1632 sage: J(expected) == x*y
1635 We can change the generator prefix::
1637 sage: RealSymmetricEJA(3, prefix='q').gens()
1638 (q0, q1, q2, q3, q4, q5)
1640 We can construct the (trivial) algebra of rank zero::
1642 sage: RealSymmetricEJA(0)
1643 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
1647 def _denormalized_basis(cls
, n
):
1649 Return a basis for the space of real symmetric n-by-n matrices.
1653 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1657 sage: set_random_seed()
1658 sage: n = ZZ.random_element(1,5)
1659 sage: B = RealSymmetricEJA._denormalized_basis(n)
1660 sage: all( M.is_symmetric() for M in B)
1664 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1668 for j
in range(i
+1):
1669 Eij
= matrix(ZZ
, n
, lambda k
,l
: k
==i
and l
==j
)
1673 Sij
= Eij
+ Eij
.transpose()
1679 def _max_random_instance_size():
1680 return 4 # Dimension 10
1683 def random_instance(cls
, **kwargs
):
1685 Return a random instance of this type of algebra.
1687 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
1688 return cls(n
, **kwargs
)
1690 def __init__(self
, n
, **kwargs
):
1691 # We know this is a valid EJA, but will double-check
1692 # if the user passes check_axioms=True.
1693 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
1695 super(RealSymmetricEJA
, self
).__init
__(self
._denormalized
_basis
(n
),
1696 self
.jordan_product
,
1697 self
.trace_inner_product
,
1700 # TODO: this could be factored out somehow, but is left here
1701 # because the MatrixEuclideanJordanAlgebra is not presently
1702 # a subclass of the FDEJA class that defines rank() and one().
1703 self
.rank
.set_cache(n
)
1704 idV
= matrix
.identity(ZZ
, self
.dimension_over_reals()*n
)
1705 self
.one
.set_cache(self(idV
))
1709 class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1711 def dimension_over_reals():
1715 def real_embed(cls
,M
):
1717 Embed the n-by-n complex matrix ``M`` into the space of real
1718 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1719 bi` to the block matrix ``[[a,b],[-b,a]]``.
1723 sage: from mjo.eja.eja_algebra import \
1724 ....: ComplexMatrixEuclideanJordanAlgebra
1728 sage: F = QuadraticField(-1, 'I')
1729 sage: x1 = F(4 - 2*i)
1730 sage: x2 = F(1 + 2*i)
1733 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1734 sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1743 Embedding is a homomorphism (isomorphism, in fact)::
1745 sage: set_random_seed()
1746 sage: n = ZZ.random_element(3)
1747 sage: F = QuadraticField(-1, 'I')
1748 sage: X = random_matrix(F, n)
1749 sage: Y = random_matrix(F, n)
1750 sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
1751 sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
1752 sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1757 super(ComplexMatrixEuclideanJordanAlgebra
,cls
).real_embed(M
)
1760 # We don't need any adjoined elements...
1761 field
= M
.base_ring().base_ring()
1765 a
= z
.list()[0] # real part, I guess
1766 b
= z
.list()[1] # imag part, I guess
1767 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1769 return matrix
.block(field
, n
, blocks
)
1773 def real_unembed(cls
,M
):
1775 The inverse of _embed_complex_matrix().
1779 sage: from mjo.eja.eja_algebra import \
1780 ....: ComplexMatrixEuclideanJordanAlgebra
1784 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1785 ....: [-2, 1, -4, 3],
1786 ....: [ 9, 10, 11, 12],
1787 ....: [-10, 9, -12, 11] ])
1788 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
1790 [ 10*I + 9 12*I + 11]
1794 Unembedding is the inverse of embedding::
1796 sage: set_random_seed()
1797 sage: F = QuadraticField(-1, 'I')
1798 sage: M = random_matrix(F, 3)
1799 sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1800 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1804 super(ComplexMatrixEuclideanJordanAlgebra
,cls
).real_unembed(M
)
1806 d
= cls
.dimension_over_reals()
1808 # If "M" was normalized, its base ring might have roots
1809 # adjoined and they can stick around after unembedding.
1810 field
= M
.base_ring()
1811 R
= PolynomialRing(field
, 'z')
1814 # Sage doesn't know how to adjoin the complex "i" (the root of
1815 # x^2 + 1) to a field in a general way. Here, we just enumerate
1816 # all of the cases that I have cared to support so far.
1818 # Sage doesn't know how to embed AA into QQbar, i.e. how
1819 # to adjoin sqrt(-1) to AA.
1821 elif not field
.is_exact():
1823 F
= field
.complex_field()
1825 # Works for QQ and... maybe some other fields.
1826 F
= field
.extension(z
**2 + 1, 'I', embedding
=CLF(-1).sqrt())
1829 # Go top-left to bottom-right (reading order), converting every
1830 # 2-by-2 block we see to a single complex element.
1832 for k
in range(n
/d
):
1833 for j
in range(n
/d
):
1834 submat
= M
[d
*k
:d
*k
+d
,d
*j
:d
*j
+d
]
1835 if submat
[0,0] != submat
[1,1]:
1836 raise ValueError('bad on-diagonal submatrix')
1837 if submat
[0,1] != -submat
[1,0]:
1838 raise ValueError('bad off-diagonal submatrix')
1839 z
= submat
[0,0] + submat
[0,1]*i
1842 return matrix(F
, n
/d
, elements
)
1845 class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra
,
1846 ComplexMatrixEuclideanJordanAlgebra
):
1848 The rank-n simple EJA consisting of complex Hermitian n-by-n
1849 matrices over the real numbers, the usual symmetric Jordan product,
1850 and the real-part-of-trace inner product. It has dimension `n^2` over
1855 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1859 In theory, our "field" can be any subfield of the reals::
1861 sage: ComplexHermitianEJA(2, field=RDF)
1862 Euclidean Jordan algebra of dimension 4 over Real Double Field
1863 sage: ComplexHermitianEJA(2, field=RR)
1864 Euclidean Jordan algebra of dimension 4 over Real Field with
1865 53 bits of precision
1869 The dimension of this algebra is `n^2`::
1871 sage: set_random_seed()
1872 sage: n_max = ComplexHermitianEJA._max_random_instance_size()
1873 sage: n = ZZ.random_element(1, n_max)
1874 sage: J = ComplexHermitianEJA(n)
1875 sage: J.dimension() == n^2
1878 The Jordan multiplication is what we think it is::
1880 sage: set_random_seed()
1881 sage: J = ComplexHermitianEJA.random_instance()
1882 sage: x,y = J.random_elements(2)
1883 sage: actual = (x*y).to_matrix()
1884 sage: X = x.to_matrix()
1885 sage: Y = y.to_matrix()
1886 sage: expected = (X*Y + Y*X)/2
1887 sage: actual == expected
1889 sage: J(expected) == x*y
1892 We can change the generator prefix::
1894 sage: ComplexHermitianEJA(2, prefix='z').gens()
1897 We can construct the (trivial) algebra of rank zero::
1899 sage: ComplexHermitianEJA(0)
1900 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
1905 def _denormalized_basis(cls
, n
):
1907 Returns a basis for the space of complex Hermitian n-by-n matrices.
1909 Why do we embed these? Basically, because all of numerical linear
1910 algebra assumes that you're working with vectors consisting of `n`
1911 entries from a field and scalars from the same field. There's no way
1912 to tell SageMath that (for example) the vectors contain complex
1913 numbers, while the scalar field is real.
1917 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1921 sage: set_random_seed()
1922 sage: n = ZZ.random_element(1,5)
1923 sage: field = QuadraticField(2, 'sqrt2')
1924 sage: B = ComplexHermitianEJA._denormalized_basis(n)
1925 sage: all( M.is_symmetric() for M in B)
1930 R
= PolynomialRing(field
, 'z')
1932 F
= field
.extension(z
**2 + 1, 'I')
1935 # This is like the symmetric case, but we need to be careful:
1937 # * We want conjugate-symmetry, not just symmetry.
1938 # * The diagonal will (as a result) be real.
1942 for j
in range(i
+1):
1943 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1945 Sij
= cls
.real_embed(Eij
)
1948 # The second one has a minus because it's conjugated.
1949 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1951 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1954 # Since we embedded these, we can drop back to the "field" that we
1955 # started with instead of the complex extension "F".
1956 return tuple( s
.change_ring(field
) for s
in S
)
1959 def __init__(self
, n
, **kwargs
):
1960 # We know this is a valid EJA, but will double-check
1961 # if the user passes check_axioms=True.
1962 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
1964 super(ComplexHermitianEJA
, self
).__init
__(self
._denormalized
_basis
(n
),
1965 self
.jordan_product
,
1966 self
.trace_inner_product
,
1968 # TODO: this could be factored out somehow, but is left here
1969 # because the MatrixEuclideanJordanAlgebra is not presently
1970 # a subclass of the FDEJA class that defines rank() and one().
1971 self
.rank
.set_cache(n
)
1972 idV
= matrix
.identity(ZZ
, self
.dimension_over_reals()*n
)
1973 self
.one
.set_cache(self(idV
))
1976 def _max_random_instance_size():
1977 return 3 # Dimension 9
1980 def random_instance(cls
, **kwargs
):
1982 Return a random instance of this type of algebra.
1984 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
1985 return cls(n
, **kwargs
)
1987 class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1989 def dimension_over_reals():
1993 def real_embed(cls
,M
):
1995 Embed the n-by-n quaternion matrix ``M`` into the space of real
1996 matrices of size 4n-by-4n by first sending each quaternion entry `z
1997 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1998 c+di],[-c + di, a-bi]]`, and then embedding those into a real
2003 sage: from mjo.eja.eja_algebra import \
2004 ....: QuaternionMatrixEuclideanJordanAlgebra
2008 sage: Q = QuaternionAlgebra(QQ,-1,-1)
2009 sage: i,j,k = Q.gens()
2010 sage: x = 1 + 2*i + 3*j + 4*k
2011 sage: M = matrix(Q, 1, [[x]])
2012 sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
2018 Embedding is a homomorphism (isomorphism, in fact)::
2020 sage: set_random_seed()
2021 sage: n = ZZ.random_element(2)
2022 sage: Q = QuaternionAlgebra(QQ,-1,-1)
2023 sage: X = random_matrix(Q, n)
2024 sage: Y = random_matrix(Q, n)
2025 sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
2026 sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
2027 sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
2032 super(QuaternionMatrixEuclideanJordanAlgebra
,cls
).real_embed(M
)
2033 quaternions
= M
.base_ring()
2036 F
= QuadraticField(-1, 'I')
2041 t
= z
.coefficient_tuple()
2046 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
2047 [-c
+ d
*i
, a
- b
*i
]])
2048 realM
= ComplexMatrixEuclideanJordanAlgebra
.real_embed(cplxM
)
2049 blocks
.append(realM
)
2051 # We should have real entries by now, so use the realest field
2052 # we've got for the return value.
2053 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
2058 def real_unembed(cls
,M
):
2060 The inverse of _embed_quaternion_matrix().
2064 sage: from mjo.eja.eja_algebra import \
2065 ....: QuaternionMatrixEuclideanJordanAlgebra
2069 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
2070 ....: [-2, 1, -4, 3],
2071 ....: [-3, 4, 1, -2],
2072 ....: [-4, -3, 2, 1]])
2073 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
2074 [1 + 2*i + 3*j + 4*k]
2078 Unembedding is the inverse of embedding::
2080 sage: set_random_seed()
2081 sage: Q = QuaternionAlgebra(QQ, -1, -1)
2082 sage: M = random_matrix(Q, 3)
2083 sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
2084 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
2088 super(QuaternionMatrixEuclideanJordanAlgebra
,cls
).real_unembed(M
)
2090 d
= cls
.dimension_over_reals()
2092 # Use the base ring of the matrix to ensure that its entries can be
2093 # multiplied by elements of the quaternion algebra.
2094 field
= M
.base_ring()
2095 Q
= QuaternionAlgebra(field
,-1,-1)
2098 # Go top-left to bottom-right (reading order), converting every
2099 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
2102 for l
in range(n
/d
):
2103 for m
in range(n
/d
):
2104 submat
= ComplexMatrixEuclideanJordanAlgebra
.real_unembed(
2105 M
[d
*l
:d
*l
+d
,d
*m
:d
*m
+d
] )
2106 if submat
[0,0] != submat
[1,1].conjugate():
2107 raise ValueError('bad on-diagonal submatrix')
2108 if submat
[0,1] != -submat
[1,0].conjugate():
2109 raise ValueError('bad off-diagonal submatrix')
2110 z
= submat
[0,0].real()
2111 z
+= submat
[0,0].imag()*i
2112 z
+= submat
[0,1].real()*j
2113 z
+= submat
[0,1].imag()*k
2116 return matrix(Q
, n
/d
, elements
)
2119 class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra
,
2120 QuaternionMatrixEuclideanJordanAlgebra
):
2122 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
2123 matrices, the usual symmetric Jordan product, and the
2124 real-part-of-trace inner product. It has dimension `2n^2 - n` over
2129 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
2133 In theory, our "field" can be any subfield of the reals::
2135 sage: QuaternionHermitianEJA(2, field=RDF)
2136 Euclidean Jordan algebra of dimension 6 over Real Double Field
2137 sage: QuaternionHermitianEJA(2, field=RR)
2138 Euclidean Jordan algebra of dimension 6 over Real Field with
2139 53 bits of precision
2143 The dimension of this algebra is `2*n^2 - n`::
2145 sage: set_random_seed()
2146 sage: n_max = QuaternionHermitianEJA._max_random_instance_size()
2147 sage: n = ZZ.random_element(1, n_max)
2148 sage: J = QuaternionHermitianEJA(n)
2149 sage: J.dimension() == 2*(n^2) - n
2152 The Jordan multiplication is what we think it is::
2154 sage: set_random_seed()
2155 sage: J = QuaternionHermitianEJA.random_instance()
2156 sage: x,y = J.random_elements(2)
2157 sage: actual = (x*y).to_matrix()
2158 sage: X = x.to_matrix()
2159 sage: Y = y.to_matrix()
2160 sage: expected = (X*Y + Y*X)/2
2161 sage: actual == expected
2163 sage: J(expected) == x*y
2166 We can change the generator prefix::
2168 sage: QuaternionHermitianEJA(2, prefix='a').gens()
2169 (a0, a1, a2, a3, a4, a5)
2171 We can construct the (trivial) algebra of rank zero::
2173 sage: QuaternionHermitianEJA(0)
2174 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
2178 def _denormalized_basis(cls
, n
):
2180 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
2182 Why do we embed these? Basically, because all of numerical
2183 linear algebra assumes that you're working with vectors consisting
2184 of `n` entries from a field and scalars from the same field. There's
2185 no way to tell SageMath that (for example) the vectors contain
2186 complex numbers, while the scalar field is real.
2190 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
2194 sage: set_random_seed()
2195 sage: n = ZZ.random_element(1,5)
2196 sage: B = QuaternionHermitianEJA._denormalized_basis(n)
2197 sage: all( M.is_symmetric() for M in B )
2202 Q
= QuaternionAlgebra(QQ
,-1,-1)
2205 # This is like the symmetric case, but we need to be careful:
2207 # * We want conjugate-symmetry, not just symmetry.
2208 # * The diagonal will (as a result) be real.
2212 for j
in range(i
+1):
2213 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
2215 Sij
= cls
.real_embed(Eij
)
2218 # The second, third, and fourth ones have a minus
2219 # because they're conjugated.
2220 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
2222 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
2224 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
2226 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
2229 # Since we embedded these, we can drop back to the "field" that we
2230 # started with instead of the quaternion algebra "Q".
2231 return tuple( s
.change_ring(field
) for s
in S
)
2234 def __init__(self
, n
, **kwargs
):
2235 # We know this is a valid EJA, but will double-check
2236 # if the user passes check_axioms=True.
2237 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2239 super(QuaternionHermitianEJA
, self
).__init
__(self
._denormalized
_basis
(n
),
2240 self
.jordan_product
,
2241 self
.trace_inner_product
,
2243 # TODO: this could be factored out somehow, but is left here
2244 # because the MatrixEuclideanJordanAlgebra is not presently
2245 # a subclass of the FDEJA class that defines rank() and one().
2246 self
.rank
.set_cache(n
)
2247 idV
= matrix
.identity(ZZ
, self
.dimension_over_reals()*n
)
2248 self
.one
.set_cache(self(idV
))
2252 def _max_random_instance_size():
2254 The maximum rank of a random QuaternionHermitianEJA.
2256 return 2 # Dimension 6
2259 def random_instance(cls
, **kwargs
):
2261 Return a random instance of this type of algebra.
2263 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2264 return cls(n
, **kwargs
)
2267 class HadamardEJA(ConcreteEuclideanJordanAlgebra
):
2269 Return the Euclidean Jordan Algebra corresponding to the set
2270 `R^n` under the Hadamard product.
2272 Note: this is nothing more than the Cartesian product of ``n``
2273 copies of the spin algebra. Once Cartesian product algebras
2274 are implemented, this can go.
2278 sage: from mjo.eja.eja_algebra import HadamardEJA
2282 This multiplication table can be verified by hand::
2284 sage: J = HadamardEJA(3)
2285 sage: e0,e1,e2 = J.gens()
2301 We can change the generator prefix::
2303 sage: HadamardEJA(3, prefix='r').gens()
2307 def __init__(self
, n
, **kwargs
):
2308 def jordan_product(x
,y
):
2310 return P(tuple( xi
*yi
for (xi
,yi
) in zip(x
,y
) ))
2311 def inner_product(x
,y
):
2312 return x
.inner_product(y
)
2314 # New defaults for keyword arguments. Don't orthonormalize
2315 # because our basis is already orthonormal with respect to our
2316 # inner-product. Don't check the axioms, because we know this
2317 # is a valid EJA... but do double-check if the user passes
2318 # check_axioms=True. Note: we DON'T override the "check_field"
2319 # default here, because the user can pass in a field!
2320 if "orthonormalize" not in kwargs
: kwargs
["orthonormalize"] = False
2321 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2324 standard_basis
= FreeModule(ZZ
, n
).basis()
2325 super(HadamardEJA
, self
).__init
__(standard_basis
,
2329 self
.rank
.set_cache(n
)
2332 self
.one
.set_cache( self
.zero() )
2334 self
.one
.set_cache( sum(self
.gens()) )
2337 def _max_random_instance_size():
2339 The maximum dimension of a random HadamardEJA.
2344 def random_instance(cls
, **kwargs
):
2346 Return a random instance of this type of algebra.
2348 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2349 return cls(n
, **kwargs
)
2352 class BilinearFormEJA(ConcreteEuclideanJordanAlgebra
):
2354 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2355 with the half-trace inner product and jordan product ``x*y =
2356 (<Bx,y>,y_bar>, x0*y_bar + y0*x_bar)`` where `B = 1 \times B22` is
2357 a symmetric positive-definite "bilinear form" matrix. Its
2358 dimension is the size of `B`, and it has rank two in dimensions
2359 larger than two. It reduces to the ``JordanSpinEJA`` when `B` is
2360 the identity matrix of order ``n``.
2362 We insist that the one-by-one upper-left identity block of `B` be
2363 passed in as well so that we can be passed a matrix of size zero
2364 to construct a trivial algebra.
2368 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
2369 ....: JordanSpinEJA)
2373 When no bilinear form is specified, the identity matrix is used,
2374 and the resulting algebra is the Jordan spin algebra::
2376 sage: B = matrix.identity(AA,3)
2377 sage: J0 = BilinearFormEJA(B)
2378 sage: J1 = JordanSpinEJA(3)
2379 sage: J0.multiplication_table() == J0.multiplication_table()
2382 An error is raised if the matrix `B` does not correspond to a
2383 positive-definite bilinear form::
2385 sage: B = matrix.random(QQ,2,3)
2386 sage: J = BilinearFormEJA(B)
2387 Traceback (most recent call last):
2389 ValueError: bilinear form is not positive-definite
2390 sage: B = matrix.zero(QQ,3)
2391 sage: J = BilinearFormEJA(B)
2392 Traceback (most recent call last):
2394 ValueError: bilinear form is not positive-definite
2398 We can create a zero-dimensional algebra::
2400 sage: B = matrix.identity(AA,0)
2401 sage: J = BilinearFormEJA(B)
2405 We can check the multiplication condition given in the Jordan, von
2406 Neumann, and Wigner paper (and also discussed on my "On the
2407 symmetry..." paper). Note that this relies heavily on the standard
2408 choice of basis, as does anything utilizing the bilinear form
2409 matrix. We opt not to orthonormalize the basis, because if we
2410 did, we would have to normalize the `s_{i}` in a similar manner::
2412 sage: set_random_seed()
2413 sage: n = ZZ.random_element(5)
2414 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
2415 sage: B11 = matrix.identity(QQ,1)
2416 sage: B22 = M.transpose()*M
2417 sage: B = block_matrix(2,2,[ [B11,0 ],
2419 sage: J = BilinearFormEJA(B, orthonormalize=False)
2420 sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
2421 sage: V = J.vector_space()
2422 sage: sis = [ J( V([0] + (M.inverse()*ei).list()).column() )
2423 ....: for ei in eis ]
2424 sage: actual = [ sis[i]*sis[j]
2425 ....: for i in range(n-1)
2426 ....: for j in range(n-1) ]
2427 sage: expected = [ J.one() if i == j else J.zero()
2428 ....: for i in range(n-1)
2429 ....: for j in range(n-1) ]
2430 sage: actual == expected
2433 def __init__(self
, B
, **kwargs
):
2434 if not B
.is_positive_definite():
2435 raise ValueError("bilinear form is not positive-definite")
2437 def inner_product(x
,y
):
2438 return (B
*x
).inner_product(y
)
2440 def jordan_product(x
,y
):
2446 z0
= inner_product(x
,y
)
2447 zbar
= y0
*xbar
+ x0
*ybar
2448 return P((z0
,) + tuple(zbar
))
2450 # We know this is a valid EJA, but will double-check
2451 # if the user passes check_axioms=True.
2452 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2455 standard_basis
= FreeModule(ZZ
, n
).basis()
2456 super(BilinearFormEJA
, self
).__init
__(standard_basis
,
2461 # The rank of this algebra is two, unless we're in a
2462 # one-dimensional ambient space (because the rank is bounded
2463 # by the ambient dimension).
2464 self
.rank
.set_cache(min(n
,2))
2467 self
.one
.set_cache( self
.zero() )
2469 self
.one
.set_cache( self
.monomial(0) )
2472 def _max_random_instance_size():
2474 The maximum dimension of a random BilinearFormEJA.
2479 def random_instance(cls
, **kwargs
):
2481 Return a random instance of this algebra.
2483 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2485 B
= matrix
.identity(ZZ
, n
)
2486 return cls(B
, **kwargs
)
2488 B11
= matrix
.identity(ZZ
, 1)
2489 M
= matrix
.random(ZZ
, n
-1)
2490 I
= matrix
.identity(ZZ
, n
-1)
2492 while alpha
.is_zero():
2493 alpha
= ZZ
.random_element().abs()
2494 B22
= M
.transpose()*M
+ alpha
*I
2496 from sage
.matrix
.special
import block_matrix
2497 B
= block_matrix(2,2, [ [B11
, ZZ(0) ],
2500 return cls(B
, **kwargs
)
2503 class JordanSpinEJA(BilinearFormEJA
):
2505 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2506 with the usual inner product and jordan product ``x*y =
2507 (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2512 sage: from mjo.eja.eja_algebra import JordanSpinEJA
2516 This multiplication table can be verified by hand::
2518 sage: J = JordanSpinEJA(4)
2519 sage: e0,e1,e2,e3 = J.gens()
2535 We can change the generator prefix::
2537 sage: JordanSpinEJA(2, prefix='B').gens()
2542 Ensure that we have the usual inner product on `R^n`::
2544 sage: set_random_seed()
2545 sage: J = JordanSpinEJA.random_instance()
2546 sage: x,y = J.random_elements(2)
2547 sage: actual = x.inner_product(y)
2548 sage: expected = x.to_vector().inner_product(y.to_vector())
2549 sage: actual == expected
2553 def __init__(self
, n
, **kwargs
):
2554 # This is a special case of the BilinearFormEJA with the
2555 # identity matrix as its bilinear form.
2556 B
= matrix
.identity(ZZ
, n
)
2558 # Don't orthonormalize because our basis is already
2559 # orthonormal with respect to our inner-product.
2560 if "orthonormalize" not in kwargs
: kwargs
["orthonormalize"] = False
2562 # But also don't pass check_field=False here, because the user
2563 # can pass in a field!
2564 super(JordanSpinEJA
, self
).__init
__(B
, **kwargs
)
2567 def _max_random_instance_size():
2569 The maximum dimension of a random JordanSpinEJA.
2574 def random_instance(cls
, **kwargs
):
2576 Return a random instance of this type of algebra.
2578 Needed here to override the implementation for ``BilinearFormEJA``.
2580 n
= ZZ
.random_element(cls
._max
_random
_instance
_size
() + 1)
2581 return cls(n
, **kwargs
)
2584 class TrivialEJA(ConcreteEuclideanJordanAlgebra
):
2586 The trivial Euclidean Jordan algebra consisting of only a zero element.
2590 sage: from mjo.eja.eja_algebra import TrivialEJA
2594 sage: J = TrivialEJA()
2601 sage: 7*J.one()*12*J.one()
2603 sage: J.one().inner_product(J.one())
2605 sage: J.one().norm()
2607 sage: J.one().subalgebra_generated_by()
2608 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
2613 def __init__(self
, **kwargs
):
2614 jordan_product
= lambda x
,y
: x
2615 inner_product
= lambda x
,y
: 0
2618 # New defaults for keyword arguments
2619 if "orthonormalize" not in kwargs
: kwargs
["orthonormalize"] = False
2620 if "check_axioms" not in kwargs
: kwargs
["check_axioms"] = False
2622 super(TrivialEJA
, self
).__init
__(basis
,
2626 # The rank is zero using my definition, namely the dimension of the
2627 # largest subalgebra generated by any element.
2628 self
.rank
.set_cache(0)
2629 self
.one
.set_cache( self
.zero() )
2632 def random_instance(cls
, **kwargs
):
2633 # We don't take a "size" argument so the superclass method is
2634 # inappropriate for us.
2635 return cls(**kwargs
)
2637 class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2639 The external (orthogonal) direct sum of two other Euclidean Jordan
2640 algebras. Essentially the Cartesian product of its two factors.
2641 Every Euclidean Jordan algebra decomposes into an orthogonal
2642 direct sum of simple Euclidean Jordan algebras, so no generality
2643 is lost by providing only this construction.
2647 sage: from mjo.eja.eja_algebra import (random_eja,
2649 ....: RealSymmetricEJA,
2654 sage: J1 = HadamardEJA(2)
2655 sage: J2 = RealSymmetricEJA(3)
2656 sage: J = DirectSumEJA(J1,J2)
2664 The external direct sum construction is only valid when the two factors
2665 have the same base ring; an error is raised otherwise::
2667 sage: set_random_seed()
2668 sage: J1 = random_eja(field=AA)
2669 sage: J2 = random_eja(field=QQ,orthonormalize=False)
2670 sage: J = DirectSumEJA(J1,J2)
2671 Traceback (most recent call last):
2673 ValueError: algebras must share the same base field
2676 def __init__(self
, J1
, J2
, **kwargs
):
2677 if J1
.base_ring() != J2
.base_ring():
2678 raise ValueError("algebras must share the same base field")
2679 field
= J1
.base_ring()
2681 self
._factors
= (J1
, J2
)
2685 V
= VectorSpace(field
, n
)
2686 mult_table
= [ [ V
.zero() for j
in range(i
+1) ]
2689 for j
in range(i
+1):
2690 p
= (J1
.monomial(i
)*J1
.monomial(j
)).to_vector()
2691 mult_table
[i
][j
] = V(p
.list() + [field
.zero()]*n2
)
2694 for j
in range(i
+1):
2695 p
= (J2
.monomial(i
)*J2
.monomial(j
)).to_vector()
2696 mult_table
[n1
+i
][n1
+j
] = V([field
.zero()]*n1
+ p
.list())
2698 # TODO: build the IP table here from the two constituent IP
2699 # matrices (it'll be block diagonal, I think).
2700 ip_table
= [ [ field
.zero() for j
in range(i
+1) ]
2702 super(DirectSumEJA
, self
).__init
__(field
,
2707 self
.rank
.set_cache(J1
.rank() + J2
.rank())
2712 Return the pair of this algebra's factors.
2716 sage: from mjo.eja.eja_algebra import (HadamardEJA,
2717 ....: JordanSpinEJA,
2722 sage: J1 = HadamardEJA(2, field=QQ)
2723 sage: J2 = JordanSpinEJA(3, field=QQ)
2724 sage: J = DirectSumEJA(J1,J2)
2726 (Euclidean Jordan algebra of dimension 2 over Rational Field,
2727 Euclidean Jordan algebra of dimension 3 over Rational Field)
2730 return self
._factors
2732 def projections(self
):
2734 Return a pair of projections onto this algebra's factors.
2738 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
2739 ....: ComplexHermitianEJA,
2744 sage: J1 = JordanSpinEJA(2)
2745 sage: J2 = ComplexHermitianEJA(2)
2746 sage: J = DirectSumEJA(J1,J2)
2747 sage: (pi_left, pi_right) = J.projections()
2748 sage: J.one().to_vector()
2750 sage: pi_left(J.one()).to_vector()
2752 sage: pi_right(J.one()).to_vector()
2756 (J1
,J2
) = self
.factors()
2759 V_basis
= self
.vector_space().basis()
2760 # Need to specify the dimensions explicitly so that we don't
2761 # wind up with a zero-by-zero matrix when we want e.g. a
2762 # zero-by-two matrix (important for composing things).
2763 P1
= matrix(self
.base_ring(), m
, m
+n
, V_basis
[:m
])
2764 P2
= matrix(self
.base_ring(), n
, m
+n
, V_basis
[m
:])
2765 pi_left
= FiniteDimensionalEuclideanJordanAlgebraOperator(self
,J1
,P1
)
2766 pi_right
= FiniteDimensionalEuclideanJordanAlgebraOperator(self
,J2
,P2
)
2767 return (pi_left
, pi_right
)
2769 def inclusions(self
):
2771 Return the pair of inclusion maps from our factors into us.
2775 sage: from mjo.eja.eja_algebra import (random_eja,
2776 ....: JordanSpinEJA,
2777 ....: RealSymmetricEJA,
2782 sage: J1 = JordanSpinEJA(3)
2783 sage: J2 = RealSymmetricEJA(2)
2784 sage: J = DirectSumEJA(J1,J2)
2785 sage: (iota_left, iota_right) = J.inclusions()
2786 sage: iota_left(J1.zero()) == J.zero()
2788 sage: iota_right(J2.zero()) == J.zero()
2790 sage: J1.one().to_vector()
2792 sage: iota_left(J1.one()).to_vector()
2794 sage: J2.one().to_vector()
2796 sage: iota_right(J2.one()).to_vector()
2798 sage: J.one().to_vector()
2803 Composing a projection with the corresponding inclusion should
2804 produce the identity map, and mismatching them should produce
2807 sage: set_random_seed()
2808 sage: J1 = random_eja()
2809 sage: J2 = random_eja()
2810 sage: J = DirectSumEJA(J1,J2)
2811 sage: (iota_left, iota_right) = J.inclusions()
2812 sage: (pi_left, pi_right) = J.projections()
2813 sage: pi_left*iota_left == J1.one().operator()
2815 sage: pi_right*iota_right == J2.one().operator()
2817 sage: (pi_left*iota_right).is_zero()
2819 sage: (pi_right*iota_left).is_zero()
2823 (J1
,J2
) = self
.factors()
2826 V_basis
= self
.vector_space().basis()
2827 # Need to specify the dimensions explicitly so that we don't
2828 # wind up with a zero-by-zero matrix when we want e.g. a
2829 # two-by-zero matrix (important for composing things).
2830 I1
= matrix
.column(self
.base_ring(), m
, m
+n
, V_basis
[:m
])
2831 I2
= matrix
.column(self
.base_ring(), n
, m
+n
, V_basis
[m
:])
2832 iota_left
= FiniteDimensionalEuclideanJordanAlgebraOperator(J1
,self
,I1
)
2833 iota_right
= FiniteDimensionalEuclideanJordanAlgebraOperator(J2
,self
,I2
)
2834 return (iota_left
, iota_right
)
2836 def inner_product(self
, x
, y
):
2838 The standard Cartesian inner-product.
2840 We project ``x`` and ``y`` onto our factors, and add up the
2841 inner-products from the subalgebras.
2846 sage: from mjo.eja.eja_algebra import (HadamardEJA,
2847 ....: QuaternionHermitianEJA,
2852 sage: J1 = HadamardEJA(3,field=QQ)
2853 sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
2854 sage: J = DirectSumEJA(J1,J2)
2859 sage: x1.inner_product(x2)
2861 sage: y1.inner_product(y2)
2863 sage: J.one().inner_product(J.one())
2867 (pi_left
, pi_right
) = self
.projections()
2873 return (x1
.inner_product(y1
) + x2
.inner_product(y2
))
2877 random_eja
= ConcreteEuclideanJordanAlgebra
.random_instance