2 Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
3 specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
4 are used in optimization, and have some additional nice methods beyond
5 what can be supported in a general Jordan Algebra.
8 from itertools
import repeat
10 from sage
.algebras
.quatalg
.quaternion_algebra
import QuaternionAlgebra
11 from sage
.categories
.magmatic_algebras
import MagmaticAlgebras
12 from sage
.combinat
.free_module
import CombinatorialFreeModule
13 from sage
.matrix
.constructor
import matrix
14 from sage
.matrix
.matrix_space
import MatrixSpace
15 from sage
.misc
.cachefunc
import cached_method
16 from sage
.misc
.lazy_import
import lazy_import
17 from sage
.misc
.prandom
import choice
18 from sage
.misc
.table
import table
19 from sage
.modules
.free_module
import FreeModule
, VectorSpace
20 from sage
.rings
.all
import (ZZ
, QQ
, AA
, QQbar
, RR
, RLF
, CLF
,
23 from mjo
.eja
.eja_element
import FiniteDimensionalEuclideanJordanAlgebraElement
24 lazy_import('mjo.eja.eja_subalgebra',
25 'FiniteDimensionalEuclideanJordanSubalgebra')
26 from mjo
.eja
.eja_utils
import _mat2vec
28 class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule
):
30 def _coerce_map_from_base_ring(self
):
32 Disable the map from the base ring into the algebra.
34 Performing a nonsense conversion like this automatically
35 is counterpedagogical. The fallback is to try the usual
36 element constructor, which should also fail.
40 sage: from mjo.eja.eja_algebra import random_eja
44 sage: set_random_seed()
45 sage: J = random_eja()
47 Traceback (most recent call last):
49 ValueError: not a naturally-represented algebra element
65 sage: from mjo.eja.eja_algebra import (JordanSpinEJA, random_eja)
69 By definition, Jordan multiplication commutes::
71 sage: set_random_seed()
72 sage: J = random_eja()
73 sage: x,y = J.random_elements(2)
79 The ``field`` we're given must be real::
81 sage: JordanSpinEJA(2,QQbar)
82 Traceback (most recent call last):
84 ValueError: field is not real
88 if not field
.is_subring(RR
):
89 # Note: this does return true for the real algebraic
90 # field, and any quadratic field where we've specified
92 raise ValueError('field is not real')
95 self
._natural
_basis
= natural_basis
98 category
= MagmaticAlgebras(field
).FiniteDimensional()
99 category
= category
.WithBasis().Unital()
101 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
103 range(len(mult_table
)),
106 self
.print_options(bracket
='')
108 # The multiplication table we're given is necessarily in terms
109 # of vectors, because we don't have an algebra yet for
110 # anything to be an element of. However, it's faster in the
111 # long run to have the multiplication table be in terms of
112 # algebra elements. We do this after calling the superclass
113 # constructor so that from_vector() knows what to do.
114 self
._multiplication
_table
= [
115 list(map(lambda x
: self
.from_vector(x
), ls
))
120 def _element_constructor_(self
, elt
):
122 Construct an element of this algebra from its natural
125 This gets called only after the parent element _call_ method
126 fails to find a coercion for the argument.
130 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
132 ....: RealSymmetricEJA)
136 The identity in `S^n` is converted to the identity in the EJA::
138 sage: J = RealSymmetricEJA(3)
139 sage: I = matrix.identity(QQ,3)
140 sage: J(I) == J.one()
143 This skew-symmetric matrix can't be represented in the EJA::
145 sage: J = RealSymmetricEJA(3)
146 sage: A = matrix(QQ,3, lambda i,j: i-j)
148 Traceback (most recent call last):
150 ArithmeticError: vector is not in free module
154 Ensure that we can convert any element of the two non-matrix
155 simple algebras (whose natural representations are their usual
156 vector representations) back and forth faithfully::
158 sage: set_random_seed()
159 sage: J = HadamardEJA.random_instance()
160 sage: x = J.random_element()
161 sage: J(x.to_vector().column()) == x
163 sage: J = JordanSpinEJA.random_instance()
164 sage: x = J.random_element()
165 sage: J(x.to_vector().column()) == x
169 msg
= "not a naturally-represented algebra element"
171 # The superclass implementation of random_element()
172 # needs to be able to coerce "0" into the algebra.
174 elif elt
in self
.base_ring():
175 # Ensure that no base ring -> algebra coercion is performed
176 # by this method. There's some stupidity in sage that would
177 # otherwise propagate to this method; for example, sage thinks
178 # that the integer 3 belongs to the space of 2-by-2 matrices.
179 raise ValueError(msg
)
181 natural_basis
= self
.natural_basis()
182 basis_space
= natural_basis
[0].matrix_space()
183 if elt
not in basis_space
:
184 raise ValueError(msg
)
186 # Thanks for nothing! Matrix spaces aren't vector spaces in
187 # Sage, so we have to figure out its natural-basis coordinates
188 # ourselves. We use the basis space's ring instead of the
189 # element's ring because the basis space might be an algebraic
190 # closure whereas the base ring of the 3-by-3 identity matrix
191 # could be QQ instead of QQbar.
192 V
= VectorSpace(basis_space
.base_ring(), elt
.nrows()*elt
.ncols())
193 W
= V
.span_of_basis( _mat2vec(s
) for s
in natural_basis
)
194 coords
= W
.coordinate_vector(_mat2vec(elt
))
195 return self
.from_vector(coords
)
200 Return a string representation of ``self``.
204 sage: from mjo.eja.eja_algebra import JordanSpinEJA
208 Ensure that it says what we think it says::
210 sage: JordanSpinEJA(2, field=AA)
211 Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
212 sage: JordanSpinEJA(3, field=RDF)
213 Euclidean Jordan algebra of dimension 3 over Real Double Field
216 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
217 return fmt
.format(self
.dimension(), self
.base_ring())
219 def product_on_basis(self
, i
, j
):
220 return self
._multiplication
_table
[i
][j
]
222 def _a_regular_element(self
):
224 Guess a regular element. Needed to compute the basis for our
225 characteristic polynomial coefficients.
229 sage: from mjo.eja.eja_algebra import random_eja
233 Ensure that this hacky method succeeds for every algebra that we
234 know how to construct::
236 sage: set_random_seed()
237 sage: J = random_eja()
238 sage: J._a_regular_element().is_regular()
243 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
244 if not z
.is_regular():
245 raise ValueError("don't know a regular element")
250 def _charpoly_basis_space(self
):
252 Return the vector space spanned by the basis used in our
253 characteristic polynomial coefficients. This is used not only to
254 compute those coefficients, but also any time we need to
255 evaluate the coefficients (like when we compute the trace or
258 z
= self
._a
_regular
_element
()
259 # Don't use the parent vector space directly here in case this
260 # happens to be a subalgebra. In that case, we would be e.g.
261 # two-dimensional but span_of_basis() would expect three
263 V
= VectorSpace(self
.base_ring(), self
.vector_space().dimension())
264 basis
= [ (z
**k
).to_vector() for k
in range(self
.rank()) ]
265 V1
= V
.span_of_basis( basis
)
266 b
= (V1
.basis() + V1
.complement().basis())
267 return V
.span_of_basis(b
)
272 def _charpoly_coeff(self
, i
):
274 Return the coefficient polynomial "a_{i}" of this algebra's
275 general characteristic polynomial.
277 Having this be a separate cached method lets us compute and
278 store the trace/determinant (a_{r-1} and a_{0} respectively)
279 separate from the entire characteristic polynomial.
281 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
282 R
= A_of_x
.base_ring()
287 # Guaranteed by theory
290 # Danger: the in-place modification is done for performance
291 # reasons (reconstructing a matrix with huge polynomial
292 # entries is slow), but I don't know how cached_method works,
293 # so it's highly possible that we're modifying some global
294 # list variable by reference, here. In other words, you
295 # probably shouldn't call this method twice on the same
296 # algebra, at the same time, in two threads
297 Ai_orig
= A_of_x
.column(i
)
298 A_of_x
.set_column(i
,xr
)
299 numerator
= A_of_x
.det()
300 A_of_x
.set_column(i
,Ai_orig
)
302 # We're relying on the theory here to ensure that each a_i is
303 # indeed back in R, and the added negative signs are to make
304 # the whole charpoly expression sum to zero.
305 return R(-numerator
/detA
)
309 def _charpoly_matrix_system(self
):
311 Compute the matrix whose entries A_ij are polynomials in
312 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
313 corresponding to `x^r` and the determinent of the matrix A =
314 [A_ij]. In other words, all of the fixed (cachable) data needed
315 to compute the coefficients of the characteristic polynomial.
320 # Turn my vector space into a module so that "vectors" can
321 # have multivatiate polynomial entries.
322 names
= tuple('X' + str(i
) for i
in range(1,n
+1))
323 R
= PolynomialRing(self
.base_ring(), names
)
325 # Using change_ring() on the parent's vector space doesn't work
326 # here because, in a subalgebra, that vector space has a basis
327 # and change_ring() tries to bring the basis along with it. And
328 # that doesn't work unless the new ring is a PID, which it usually
332 # Now let x = (X1,X2,...,Xn) be the vector whose entries are
336 # And figure out the "left multiplication by x" matrix in
339 monomial_matrices
= [ self
.monomial(i
).operator().matrix()
340 for i
in range(n
) ] # don't recompute these!
342 ek
= self
.monomial(k
).to_vector()
344 sum( x
[i
]*(monomial_matrices
[i
]*ek
)
345 for i
in range(n
) ) )
346 Lx
= matrix
.column(R
, lmbx_cols
)
348 # Now we can compute powers of x "symbolically"
349 x_powers
= [self
.one().to_vector(), x
]
350 for d
in range(2, r
+1):
351 x_powers
.append( Lx
*(x_powers
[-1]) )
353 idmat
= matrix
.identity(R
, n
)
355 W
= self
._charpoly
_basis
_space
()
356 W
= W
.change_ring(R
.fraction_field())
358 # Starting with the standard coordinates x = (X1,X2,...,Xn)
359 # and then converting the entries to W-coordinates allows us
360 # to pass in the standard coordinates to the charpoly and get
361 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
364 # W.coordinates(x^2) eval'd at (standard z-coords)
368 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
370 # We want the middle equivalent thing in our matrix, but use
371 # the first equivalent thing instead so that we can pass in
372 # standard coordinates.
373 x_powers
= [ W
.coordinate_vector(xp
) for xp
in x_powers
]
374 l2
= [idmat
.column(k
-1) for k
in range(r
+1, n
+1)]
375 A_of_x
= matrix
.column(R
, n
, (x_powers
[:r
] + l2
))
376 return (A_of_x
, x
, x_powers
[r
], A_of_x
.det())
380 def characteristic_polynomial(self
):
382 Return a characteristic polynomial that works for all elements
385 The resulting polynomial has `n+1` variables, where `n` is the
386 dimension of this algebra. The first `n` variables correspond to
387 the coordinates of an algebra element: when evaluated at the
388 coordinates of an algebra element with respect to a certain
389 basis, the result is a univariate polynomial (in the one
390 remaining variable ``t``), namely the characteristic polynomial
395 sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA
399 The characteristic polynomial in the spin algebra is given in
400 Alizadeh, Example 11.11::
402 sage: J = JordanSpinEJA(3)
403 sage: p = J.characteristic_polynomial(); p
404 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
405 sage: xvec = J.one().to_vector()
409 By definition, the characteristic polynomial is a monic
410 degree-zero polynomial in a rank-zero algebra. Note that
411 Cayley-Hamilton is indeed satisfied since the polynomial
412 ``1`` evaluates to the identity element of the algebra on
415 sage: J = TrivialEJA()
416 sage: J.characteristic_polynomial()
423 # The list of coefficient polynomials a_0, a_1, a_2, ..., a_n.
424 a
= [ self
._charpoly
_coeff
(i
) for i
in range(r
+1) ]
426 # We go to a bit of trouble here to reorder the
427 # indeterminates, so that it's easier to evaluate the
428 # characteristic polynomial at x's coordinates and get back
429 # something in terms of t, which is what we want.
431 S
= PolynomialRing(self
.base_ring(),'t')
433 S
= PolynomialRing(S
, R
.variable_names())
436 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
439 def inner_product(self
, x
, y
):
441 The inner product associated with this Euclidean Jordan algebra.
443 Defaults to the trace inner product, but can be overridden by
444 subclasses if they are sure that the necessary properties are
449 sage: from mjo.eja.eja_algebra import random_eja
453 Our inner product is "associative," which means the following for
454 a symmetric bilinear form::
456 sage: set_random_seed()
457 sage: J = random_eja()
458 sage: x,y,z = J.random_elements(3)
459 sage: (x*y).inner_product(z) == y.inner_product(x*z)
463 X
= x
.natural_representation()
464 Y
= y
.natural_representation()
465 return self
.natural_inner_product(X
,Y
)
468 def is_trivial(self
):
470 Return whether or not this algebra is trivial.
472 A trivial algebra contains only the zero element.
476 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
481 sage: J = ComplexHermitianEJA(3)
487 sage: J = TrivialEJA()
492 return self
.dimension() == 0
495 def multiplication_table(self
):
497 Return a visual representation of this algebra's multiplication
498 table (on basis elements).
502 sage: from mjo.eja.eja_algebra import JordanSpinEJA
506 sage: J = JordanSpinEJA(4)
507 sage: J.multiplication_table()
508 +----++----+----+----+----+
509 | * || e0 | e1 | e2 | e3 |
510 +====++====+====+====+====+
511 | e0 || e0 | e1 | e2 | e3 |
512 +----++----+----+----+----+
513 | e1 || e1 | e0 | 0 | 0 |
514 +----++----+----+----+----+
515 | e2 || e2 | 0 | e0 | 0 |
516 +----++----+----+----+----+
517 | e3 || e3 | 0 | 0 | e0 |
518 +----++----+----+----+----+
521 M
= list(self
._multiplication
_table
) # copy
522 for i
in range(len(M
)):
523 # M had better be "square"
524 M
[i
] = [self
.monomial(i
)] + M
[i
]
525 M
= [["*"] + list(self
.gens())] + M
526 return table(M
, header_row
=True, header_column
=True, frame
=True)
529 def natural_basis(self
):
531 Return a more-natural representation of this algebra's basis.
533 Every finite-dimensional Euclidean Jordan Algebra is a direct
534 sum of five simple algebras, four of which comprise Hermitian
535 matrices. This method returns the original "natural" basis
536 for our underlying vector space. (Typically, the natural basis
537 is used to construct the multiplication table in the first place.)
539 Note that this will always return a matrix. The standard basis
540 in `R^n` will be returned as `n`-by-`1` column matrices.
544 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
545 ....: RealSymmetricEJA)
549 sage: J = RealSymmetricEJA(2)
551 Finite family {0: e0, 1: e1, 2: e2}
552 sage: J.natural_basis()
554 [1 0] [ 0 0.7071067811865475?] [0 0]
555 [0 0], [0.7071067811865475? 0], [0 1]
560 sage: J = JordanSpinEJA(2)
562 Finite family {0: e0, 1: e1}
563 sage: J.natural_basis()
570 if self
._natural
_basis
is None:
571 M
= self
.natural_basis_space()
572 return tuple( M(b
.to_vector()) for b
in self
.basis() )
574 return self
._natural
_basis
577 def natural_basis_space(self
):
579 Return the matrix space in which this algebra's natural basis
582 if self
._natural
_basis
is None or len(self
._natural
_basis
) == 0:
583 return MatrixSpace(self
.base_ring(), self
.dimension(), 1)
585 return self
._natural
_basis
[0].matrix_space()
589 def natural_inner_product(X
,Y
):
591 Compute the inner product of two naturally-represented elements.
593 For example in the real symmetric matrix EJA, this will compute
594 the trace inner-product of two n-by-n symmetric matrices. The
595 default should work for the real cartesian product EJA, the
596 Jordan spin EJA, and the real symmetric matrices. The others
597 will have to be overridden.
599 return (X
.conjugate_transpose()*Y
).trace()
605 Return the unit element of this algebra.
609 sage: from mjo.eja.eja_algebra import (HadamardEJA,
614 sage: J = HadamardEJA(5)
616 e0 + e1 + e2 + e3 + e4
620 The identity element acts like the identity::
622 sage: set_random_seed()
623 sage: J = random_eja()
624 sage: x = J.random_element()
625 sage: J.one()*x == x and x*J.one() == x
628 The matrix of the unit element's operator is the identity::
630 sage: set_random_seed()
631 sage: J = random_eja()
632 sage: actual = J.one().operator().matrix()
633 sage: expected = matrix.identity(J.base_ring(), J.dimension())
634 sage: actual == expected
638 # We can brute-force compute the matrices of the operators
639 # that correspond to the basis elements of this algebra.
640 # If some linear combination of those basis elements is the
641 # algebra identity, then the same linear combination of
642 # their matrices has to be the identity matrix.
644 # Of course, matrices aren't vectors in sage, so we have to
645 # appeal to the "long vectors" isometry.
646 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
648 # Now we use basis linear algebra to find the coefficients,
649 # of the matrices-as-vectors-linear-combination, which should
650 # work for the original algebra basis too.
651 A
= matrix
.column(self
.base_ring(), oper_vecs
)
653 # We used the isometry on the left-hand side already, but we
654 # still need to do it for the right-hand side. Recall that we
655 # wanted something that summed to the identity matrix.
656 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
658 # Now if there's an identity element in the algebra, this should work.
659 coeffs
= A
.solve_right(b
)
660 return self
.linear_combination(zip(self
.gens(), coeffs
))
663 def peirce_decomposition(self
, c
):
665 The Peirce decomposition of this algebra relative to the
668 In the future, this can be extended to a complete system of
669 orthogonal idempotents.
673 - ``c`` -- an idempotent of this algebra.
677 A triple (J0, J5, J1) containing two subalgebras and one subspace
680 - ``J0`` -- the algebra on the eigenspace of ``c.operator()``
681 corresponding to the eigenvalue zero.
683 - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()``
684 corresponding to the eigenvalue one-half.
686 - ``J1`` -- the algebra on the eigenspace of ``c.operator()``
687 corresponding to the eigenvalue one.
689 These are the only possible eigenspaces for that operator, and this
690 algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are
691 orthogonal, and are subalgebras of this algebra with the appropriate
696 sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA
700 The canonical example comes from the symmetric matrices, which
701 decompose into diagonal and off-diagonal parts::
703 sage: J = RealSymmetricEJA(3)
704 sage: C = matrix(QQ, [ [1,0,0],
708 sage: J0,J5,J1 = J.peirce_decomposition(c)
710 Euclidean Jordan algebra of dimension 1...
712 Vector space of degree 6 and dimension 2...
714 Euclidean Jordan algebra of dimension 3...
718 Every algebra decomposes trivially with respect to its identity
721 sage: set_random_seed()
722 sage: J = random_eja()
723 sage: J0,J5,J1 = J.peirce_decomposition(J.one())
724 sage: J0.dimension() == 0 and J5.dimension() == 0
726 sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
729 The identity elements in the two subalgebras are the
730 projections onto their respective subspaces of the
731 superalgebra's identity element::
733 sage: set_random_seed()
734 sage: J = random_eja()
735 sage: x = J.random_element()
736 sage: if not J.is_trivial():
737 ....: while x.is_nilpotent():
738 ....: x = J.random_element()
739 sage: c = x.subalgebra_idempotent()
740 sage: J0,J5,J1 = J.peirce_decomposition(c)
741 sage: J1(c) == J1.one()
743 sage: J0(J.one() - c) == J0.one()
747 if not c
.is_idempotent():
748 raise ValueError("element is not idempotent: %s" % c
)
750 # Default these to what they should be if they turn out to be
751 # trivial, because eigenspaces_left() won't return eigenvalues
752 # corresponding to trivial spaces (e.g. it returns only the
753 # eigenspace corresponding to lambda=1 if you take the
754 # decomposition relative to the identity element).
755 trivial
= FiniteDimensionalEuclideanJordanSubalgebra(self
, ())
756 J0
= trivial
# eigenvalue zero
757 J5
= VectorSpace(self
.base_ring(), 0) # eigenvalue one-half
758 J1
= trivial
# eigenvalue one
760 for (eigval
, eigspace
) in c
.operator().matrix().right_eigenspaces():
761 if eigval
== ~
(self
.base_ring()(2)):
764 gens
= tuple( self
.from_vector(b
) for b
in eigspace
.basis() )
765 subalg
= FiniteDimensionalEuclideanJordanSubalgebra(self
, gens
)
771 raise ValueError("unexpected eigenvalue: %s" % eigval
)
776 def random_elements(self
, count
):
778 Return ``count`` random elements as a tuple.
782 sage: from mjo.eja.eja_algebra import JordanSpinEJA
786 sage: J = JordanSpinEJA(3)
787 sage: x,y,z = J.random_elements(3)
788 sage: all( [ x in J, y in J, z in J ])
790 sage: len( J.random_elements(10) ) == 10
794 return tuple( self
.random_element() for idx
in range(count
) )
797 def _rank_computation(self
):
799 Compute the rank of this algebra using highly suspicious voodoo.
803 We first compute the basis representation of the operator L_x
804 using polynomial indeterminates are placeholders for the
805 coordinates of "x", which is arbitrary. We then use that
806 matrix to compute the (polynomial) entries of x^0, x^1, ...,
807 x^d,... for increasing values of "d", starting at zero. The
808 idea is that. If we also add "coefficient variables" a_0,
809 a_1,... to the ring, we can form the linear combination
810 a_0*x^0 + ... + a_d*x^d = 0, and ask what dimension the
811 solution space has as an affine variety. When "d" is smaller
812 than the rank, we expect that dimension to be the number of
813 coordinates of "x", since we can set *those* to whatever we
814 want, but linear independence forces the coefficients a_i to
815 be zero. Eventually, when "d" passes the rank, the dimension
816 of the solution space begins to grow, because we can *still*
817 set the coordinates of "x" arbitrarily, but now there are some
818 coefficients that make the sum zero as well. So, when the
819 dimension of the variety jumps, we return the corresponding
820 "d" as the rank of the algebra. This appears to work.
824 sage: from mjo.eja.eja_algebra import (HadamardEJA,
826 ....: RealSymmetricEJA,
827 ....: ComplexHermitianEJA,
828 ....: QuaternionHermitianEJA)
832 sage: J = HadamardEJA(5)
833 sage: J._rank_computation() == J.rank()
835 sage: J = JordanSpinEJA(5)
836 sage: J._rank_computation() == J.rank()
838 sage: J = RealSymmetricEJA(4)
839 sage: J._rank_computation() == J.rank()
841 sage: J = ComplexHermitianEJA(3)
842 sage: J._rank_computation() == J.rank()
844 sage: J = QuaternionHermitianEJA(2)
845 sage: J._rank_computation() == J.rank()
850 var_names
= [ "X" + str(z
) for z
in range(1,n
+1) ]
852 ideal_dim
= len(var_names
)
854 # From a result in my book, these are the entries of the
855 # basis representation of L_x.
856 return sum( vars[d
+k
]*self
.monomial(k
).operator().matrix()[i
,j
]
859 while ideal_dim
== len(var_names
):
860 coeff_names
= [ "a" + str(z
) for z
in range(d
) ]
861 R
= PolynomialRing(self
.base_ring(), coeff_names
+ var_names
)
863 L_x
= matrix(R
, n
, n
, L_x_i_j
)
864 x_powers
= [ vars[k
]*(L_x
**k
)*self
.one().to_vector()
866 eqs
= [ sum(x_powers
[k
][j
] for k
in range(d
)) for j
in range(n
) ]
867 ideal_dim
= R
.ideal(eqs
).dimension()
870 # Subtract one because we increment one too many times, and
871 # subtract another one because "d" is one greater than the
872 # answer anyway; when d=3, we go up to x^2.
877 Return the rank of this EJA.
881 The author knows of no algorithm to compute the rank of an EJA
882 where only the multiplication table is known. In lieu of one, we
883 require the rank to be specified when the algebra is created,
884 and simply pass along that number here.
888 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
889 ....: RealSymmetricEJA,
890 ....: ComplexHermitianEJA,
891 ....: QuaternionHermitianEJA,
896 The rank of the Jordan spin algebra is always two::
898 sage: JordanSpinEJA(2).rank()
900 sage: JordanSpinEJA(3).rank()
902 sage: JordanSpinEJA(4).rank()
905 The rank of the `n`-by-`n` Hermitian real, complex, or
906 quaternion matrices is `n`::
908 sage: RealSymmetricEJA(4).rank()
910 sage: ComplexHermitianEJA(3).rank()
912 sage: QuaternionHermitianEJA(2).rank()
917 Ensure that every EJA that we know how to construct has a
918 positive integer rank, unless the algebra is trivial in
919 which case its rank will be zero::
921 sage: set_random_seed()
922 sage: J = random_eja()
926 sage: r > 0 or (r == 0 and J.is_trivial())
933 def vector_space(self
):
935 Return the vector space that underlies this algebra.
939 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
943 sage: J = RealSymmetricEJA(2)
944 sage: J.vector_space()
945 Vector space of dimension 3 over...
948 return self
.zero().to_vector().parent().ambient_vector_space()
951 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
954 class KnownRankEJA(object):
956 A class for algebras that we actually know we can construct. The
957 main issue is that, for most of our methods to make sense, we need
958 to know the rank of our algebra. Thus we can't simply generate a
959 "random" algebra, or even check that a given basis and product
960 satisfy the axioms; because even if everything looks OK, we wouldn't
961 know the rank we need to actuallty build the thing.
963 Not really a subclass of FDEJA because doing that causes method
964 resolution errors, e.g.
966 TypeError: Error when calling the metaclass bases
967 Cannot create a consistent method resolution
968 order (MRO) for bases FiniteDimensionalEuclideanJordanAlgebra,
973 def _max_test_case_size():
975 Return an integer "size" that is an upper bound on the size of
976 this algebra when it is used in a random test
977 case. Unfortunately, the term "size" is quite vague -- when
978 dealing with `R^n` under either the Hadamard or Jordan spin
979 product, the "size" refers to the dimension `n`. When dealing
980 with a matrix algebra (real symmetric or complex/quaternion
981 Hermitian), it refers to the size of the matrix, which is
982 far less than the dimension of the underlying vector space.
984 We default to five in this class, which is safe in `R^n`. The
985 matrix algebra subclasses (or any class where the "size" is
986 interpreted to be far less than the dimension) should override
987 with a smaller number.
992 def random_instance(cls
, field
=AA
, **kwargs
):
994 Return a random instance of this type of algebra.
996 Beware, this will crash for "most instances" because the
997 constructor below looks wrong.
999 if cls
is TrivialEJA
:
1000 # The TrivialEJA class doesn't take an "n" argument because
1004 n
= ZZ
.random_element(cls
._max
_test
_case
_size
()) + 1
1005 return cls(n
, field
, **kwargs
)
1008 class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra
, KnownRankEJA
):
1010 Return the Euclidean Jordan Algebra corresponding to the set
1011 `R^n` under the Hadamard product.
1013 Note: this is nothing more than the Cartesian product of ``n``
1014 copies of the spin algebra. Once Cartesian product algebras
1015 are implemented, this can go.
1019 sage: from mjo.eja.eja_algebra import HadamardEJA
1023 This multiplication table can be verified by hand::
1025 sage: J = HadamardEJA(3)
1026 sage: e0,e1,e2 = J.gens()
1042 We can change the generator prefix::
1044 sage: HadamardEJA(3, prefix='r').gens()
1048 def __init__(self
, n
, field
=AA
, **kwargs
):
1049 V
= VectorSpace(field
, n
)
1050 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in range(n
) ]
1053 fdeja
= super(HadamardEJA
, self
)
1054 return fdeja
.__init
__(field
, mult_table
, rank
=n
, **kwargs
)
1056 def inner_product(self
, x
, y
):
1058 Faster to reimplement than to use natural representations.
1062 sage: from mjo.eja.eja_algebra import HadamardEJA
1066 Ensure that this is the usual inner product for the algebras
1069 sage: set_random_seed()
1070 sage: J = HadamardEJA.random_instance()
1071 sage: x,y = J.random_elements(2)
1072 sage: X = x.natural_representation()
1073 sage: Y = y.natural_representation()
1074 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
1078 return x
.to_vector().inner_product(y
.to_vector())
1081 def random_eja(field
=AA
, nontrivial
=False):
1083 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1087 sage: from mjo.eja.eja_algebra import random_eja
1092 Euclidean Jordan algebra of dimension...
1095 eja_classes
= KnownRankEJA
.__subclasses
__()
1097 eja_classes
.remove(TrivialEJA
)
1098 classname
= choice(eja_classes
)
1099 return classname
.random_instance(field
=field
)
1106 class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
1108 def _max_test_case_size():
1109 # Play it safe, since this will be squared and the underlying
1110 # field can have dimension 4 (quaternions) too.
1113 def __init__(self
, field
, basis
, rank
, normalize_basis
=True, **kwargs
):
1115 Compared to the superclass constructor, we take a basis instead of
1116 a multiplication table because the latter can be computed in terms
1117 of the former when the product is known (like it is here).
1119 # Used in this class's fast _charpoly_coeff() override.
1120 self
._basis
_normalizers
= None
1122 # We're going to loop through this a few times, so now's a good
1123 # time to ensure that it isn't a generator expression.
1124 basis
= tuple(basis
)
1126 if rank
> 1 and normalize_basis
:
1127 # We'll need sqrt(2) to normalize the basis, and this
1128 # winds up in the multiplication table, so the whole
1129 # algebra needs to be over the field extension.
1130 R
= PolynomialRing(field
, 'z')
1133 if p
.is_irreducible():
1134 field
= field
.extension(p
, 'sqrt2', embedding
=RLF(2).sqrt())
1135 basis
= tuple( s
.change_ring(field
) for s
in basis
)
1136 self
._basis
_normalizers
= tuple(
1137 ~
(self
.natural_inner_product(s
,s
).sqrt()) for s
in basis
)
1138 basis
= tuple(s
*c
for (s
,c
) in zip(basis
,self
._basis
_normalizers
))
1140 Qs
= self
.multiplication_table_from_matrix_basis(basis
)
1142 fdeja
= super(MatrixEuclideanJordanAlgebra
, self
)
1143 return fdeja
.__init
__(field
,
1146 natural_basis
=basis
,
1151 def _charpoly_coeff(self
, i
):
1153 Override the parent method with something that tries to compute
1154 over a faster (non-extension) field.
1156 if self
._basis
_normalizers
is None:
1157 # We didn't normalize, so assume that the basis we started
1158 # with had entries in a nice field.
1159 return super(MatrixEuclideanJordanAlgebra
, self
)._charpoly
_coeff
(i
)
1161 basis
= ( (b
/n
) for (b
,n
) in zip(self
.natural_basis(),
1162 self
._basis
_normalizers
) )
1164 # Do this over the rationals and convert back at the end.
1165 J
= MatrixEuclideanJordanAlgebra(QQ
,
1168 normalize_basis
=False)
1169 (_
,x
,_
,_
) = J
._charpoly
_matrix
_system
()
1170 p
= J
._charpoly
_coeff
(i
)
1171 # p might be missing some vars, have to substitute "optionally"
1172 pairs
= zip(x
.base_ring().gens(), self
._basis
_normalizers
)
1173 substitutions
= { v: v*c for (v,c) in pairs }
1174 result
= p
.subs(substitutions
)
1176 # The result of "subs" can be either a coefficient-ring
1177 # element or a polynomial. Gotta handle both cases.
1179 return self
.base_ring()(result
)
1181 return result
.change_ring(self
.base_ring())
1185 def multiplication_table_from_matrix_basis(basis
):
1187 At least three of the five simple Euclidean Jordan algebras have the
1188 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1189 multiplication on the right is matrix multiplication. Given a basis
1190 for the underlying matrix space, this function returns a
1191 multiplication table (obtained by looping through the basis
1192 elements) for an algebra of those matrices.
1194 # In S^2, for example, we nominally have four coordinates even
1195 # though the space is of dimension three only. The vector space V
1196 # is supposed to hold the entire long vector, and the subspace W
1197 # of V will be spanned by the vectors that arise from symmetric
1198 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1199 field
= basis
[0].base_ring()
1200 dimension
= basis
[0].nrows()
1202 V
= VectorSpace(field
, dimension
**2)
1203 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
1205 mult_table
= [[W
.zero() for j
in range(n
)] for i
in range(n
)]
1208 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
1209 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
1217 Embed the matrix ``M`` into a space of real matrices.
1219 The matrix ``M`` can have entries in any field at the moment:
1220 the real numbers, complex numbers, or quaternions. And although
1221 they are not a field, we can probably support octonions at some
1222 point, too. This function returns a real matrix that "acts like"
1223 the original with respect to matrix multiplication; i.e.
1225 real_embed(M*N) = real_embed(M)*real_embed(N)
1228 raise NotImplementedError
1232 def real_unembed(M
):
1234 The inverse of :meth:`real_embed`.
1236 raise NotImplementedError
1240 def natural_inner_product(cls
,X
,Y
):
1241 Xu
= cls
.real_unembed(X
)
1242 Yu
= cls
.real_unembed(Y
)
1243 tr
= (Xu
*Yu
).trace()
1246 # It's real already.
1249 # Otherwise, try the thing that works for complex numbers; and
1250 # if that doesn't work, the thing that works for quaternions.
1252 return tr
.vector()[0] # real part, imag part is index 1
1253 except AttributeError:
1254 # A quaternions doesn't have a vector() method, but does
1255 # have coefficient_tuple() method that returns the
1256 # coefficients of 1, i, j, and k -- in that order.
1257 return tr
.coefficient_tuple()[0]
1260 class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1264 The identity function, for embedding real matrices into real
1270 def real_unembed(M
):
1272 The identity function, for unembedding real matrices from real
1278 class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra
, KnownRankEJA
):
1280 The rank-n simple EJA consisting of real symmetric n-by-n
1281 matrices, the usual symmetric Jordan product, and the trace inner
1282 product. It has dimension `(n^2 + n)/2` over the reals.
1286 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1290 sage: J = RealSymmetricEJA(2)
1291 sage: e0, e1, e2 = J.gens()
1299 In theory, our "field" can be any subfield of the reals::
1301 sage: RealSymmetricEJA(2, RDF)
1302 Euclidean Jordan algebra of dimension 3 over Real Double Field
1303 sage: RealSymmetricEJA(2, RR)
1304 Euclidean Jordan algebra of dimension 3 over Real Field with
1305 53 bits of precision
1309 The dimension of this algebra is `(n^2 + n) / 2`::
1311 sage: set_random_seed()
1312 sage: n_max = RealSymmetricEJA._max_test_case_size()
1313 sage: n = ZZ.random_element(1, n_max)
1314 sage: J = RealSymmetricEJA(n)
1315 sage: J.dimension() == (n^2 + n)/2
1318 The Jordan multiplication is what we think it is::
1320 sage: set_random_seed()
1321 sage: J = RealSymmetricEJA.random_instance()
1322 sage: x,y = J.random_elements(2)
1323 sage: actual = (x*y).natural_representation()
1324 sage: X = x.natural_representation()
1325 sage: Y = y.natural_representation()
1326 sage: expected = (X*Y + Y*X)/2
1327 sage: actual == expected
1329 sage: J(expected) == x*y
1332 We can change the generator prefix::
1334 sage: RealSymmetricEJA(3, prefix='q').gens()
1335 (q0, q1, q2, q3, q4, q5)
1337 Our natural basis is normalized with respect to the natural inner
1338 product unless we specify otherwise::
1340 sage: set_random_seed()
1341 sage: J = RealSymmetricEJA.random_instance()
1342 sage: all( b.norm() == 1 for b in J.gens() )
1345 Since our natural basis is normalized with respect to the natural
1346 inner product, and since we know that this algebra is an EJA, any
1347 left-multiplication operator's matrix will be symmetric because
1348 natural->EJA basis representation is an isometry and within the EJA
1349 the operator is self-adjoint by the Jordan axiom::
1351 sage: set_random_seed()
1352 sage: x = RealSymmetricEJA.random_instance().random_element()
1353 sage: x.operator().matrix().is_symmetric()
1358 def _denormalized_basis(cls
, n
, field
):
1360 Return a basis for the space of real symmetric n-by-n matrices.
1364 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1368 sage: set_random_seed()
1369 sage: n = ZZ.random_element(1,5)
1370 sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
1371 sage: all( M.is_symmetric() for M in B)
1375 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1379 for j
in range(i
+1):
1380 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1384 Sij
= Eij
+ Eij
.transpose()
1390 def _max_test_case_size():
1391 return 4 # Dimension 10
1394 def __init__(self
, n
, field
=AA
, **kwargs
):
1395 basis
= self
._denormalized
_basis
(n
, field
)
1396 super(RealSymmetricEJA
, self
).__init
__(field
, basis
, n
, **kwargs
)
1399 class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1403 Embed the n-by-n complex matrix ``M`` into the space of real
1404 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1405 bi` to the block matrix ``[[a,b],[-b,a]]``.
1409 sage: from mjo.eja.eja_algebra import \
1410 ....: ComplexMatrixEuclideanJordanAlgebra
1414 sage: F = QuadraticField(-1, 'I')
1415 sage: x1 = F(4 - 2*i)
1416 sage: x2 = F(1 + 2*i)
1419 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1420 sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1429 Embedding is a homomorphism (isomorphism, in fact)::
1431 sage: set_random_seed()
1432 sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
1433 sage: n = ZZ.random_element(n_max)
1434 sage: F = QuadraticField(-1, 'I')
1435 sage: X = random_matrix(F, n)
1436 sage: Y = random_matrix(F, n)
1437 sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
1438 sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
1439 sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1446 raise ValueError("the matrix 'M' must be square")
1448 # We don't need any adjoined elements...
1449 field
= M
.base_ring().base_ring()
1453 a
= z
.list()[0] # real part, I guess
1454 b
= z
.list()[1] # imag part, I guess
1455 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1457 return matrix
.block(field
, n
, blocks
)
1461 def real_unembed(M
):
1463 The inverse of _embed_complex_matrix().
1467 sage: from mjo.eja.eja_algebra import \
1468 ....: ComplexMatrixEuclideanJordanAlgebra
1472 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1473 ....: [-2, 1, -4, 3],
1474 ....: [ 9, 10, 11, 12],
1475 ....: [-10, 9, -12, 11] ])
1476 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
1478 [ 10*I + 9 12*I + 11]
1482 Unembedding is the inverse of embedding::
1484 sage: set_random_seed()
1485 sage: F = QuadraticField(-1, 'I')
1486 sage: M = random_matrix(F, 3)
1487 sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1488 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1494 raise ValueError("the matrix 'M' must be square")
1495 if not n
.mod(2).is_zero():
1496 raise ValueError("the matrix 'M' must be a complex embedding")
1498 # If "M" was normalized, its base ring might have roots
1499 # adjoined and they can stick around after unembedding.
1500 field
= M
.base_ring()
1501 R
= PolynomialRing(field
, 'z')
1504 # Sage doesn't know how to embed AA into QQbar, i.e. how
1505 # to adjoin sqrt(-1) to AA.
1508 F
= field
.extension(z
**2 + 1, 'I', embedding
=CLF(-1).sqrt())
1511 # Go top-left to bottom-right (reading order), converting every
1512 # 2-by-2 block we see to a single complex element.
1514 for k
in range(n
/2):
1515 for j
in range(n
/2):
1516 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1517 if submat
[0,0] != submat
[1,1]:
1518 raise ValueError('bad on-diagonal submatrix')
1519 if submat
[0,1] != -submat
[1,0]:
1520 raise ValueError('bad off-diagonal submatrix')
1521 z
= submat
[0,0] + submat
[0,1]*i
1524 return matrix(F
, n
/2, elements
)
1528 def natural_inner_product(cls
,X
,Y
):
1530 Compute a natural inner product in this algebra directly from
1535 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1539 This gives the same answer as the slow, default method implemented
1540 in :class:`MatrixEuclideanJordanAlgebra`::
1542 sage: set_random_seed()
1543 sage: J = ComplexHermitianEJA.random_instance()
1544 sage: x,y = J.random_elements(2)
1545 sage: Xe = x.natural_representation()
1546 sage: Ye = y.natural_representation()
1547 sage: X = ComplexHermitianEJA.real_unembed(Xe)
1548 sage: Y = ComplexHermitianEJA.real_unembed(Ye)
1549 sage: expected = (X*Y).trace().real()
1550 sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
1551 sage: actual == expected
1555 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/2
1558 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra
, KnownRankEJA
):
1560 The rank-n simple EJA consisting of complex Hermitian n-by-n
1561 matrices over the real numbers, the usual symmetric Jordan product,
1562 and the real-part-of-trace inner product. It has dimension `n^2` over
1567 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1571 In theory, our "field" can be any subfield of the reals::
1573 sage: ComplexHermitianEJA(2, RDF)
1574 Euclidean Jordan algebra of dimension 4 over Real Double Field
1575 sage: ComplexHermitianEJA(2, RR)
1576 Euclidean Jordan algebra of dimension 4 over Real Field with
1577 53 bits of precision
1581 The dimension of this algebra is `n^2`::
1583 sage: set_random_seed()
1584 sage: n_max = ComplexHermitianEJA._max_test_case_size()
1585 sage: n = ZZ.random_element(1, n_max)
1586 sage: J = ComplexHermitianEJA(n)
1587 sage: J.dimension() == n^2
1590 The Jordan multiplication is what we think it is::
1592 sage: set_random_seed()
1593 sage: J = ComplexHermitianEJA.random_instance()
1594 sage: x,y = J.random_elements(2)
1595 sage: actual = (x*y).natural_representation()
1596 sage: X = x.natural_representation()
1597 sage: Y = y.natural_representation()
1598 sage: expected = (X*Y + Y*X)/2
1599 sage: actual == expected
1601 sage: J(expected) == x*y
1604 We can change the generator prefix::
1606 sage: ComplexHermitianEJA(2, prefix='z').gens()
1609 Our natural basis is normalized with respect to the natural inner
1610 product unless we specify otherwise::
1612 sage: set_random_seed()
1613 sage: J = ComplexHermitianEJA.random_instance()
1614 sage: all( b.norm() == 1 for b in J.gens() )
1617 Since our natural basis is normalized with respect to the natural
1618 inner product, and since we know that this algebra is an EJA, any
1619 left-multiplication operator's matrix will be symmetric because
1620 natural->EJA basis representation is an isometry and within the EJA
1621 the operator is self-adjoint by the Jordan axiom::
1623 sage: set_random_seed()
1624 sage: x = ComplexHermitianEJA.random_instance().random_element()
1625 sage: x.operator().matrix().is_symmetric()
1631 def _denormalized_basis(cls
, n
, field
):
1633 Returns a basis for the space of complex Hermitian n-by-n matrices.
1635 Why do we embed these? Basically, because all of numerical linear
1636 algebra assumes that you're working with vectors consisting of `n`
1637 entries from a field and scalars from the same field. There's no way
1638 to tell SageMath that (for example) the vectors contain complex
1639 numbers, while the scalar field is real.
1643 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1647 sage: set_random_seed()
1648 sage: n = ZZ.random_element(1,5)
1649 sage: field = QuadraticField(2, 'sqrt2')
1650 sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
1651 sage: all( M.is_symmetric() for M in B)
1655 R
= PolynomialRing(field
, 'z')
1657 F
= field
.extension(z
**2 + 1, 'I')
1660 # This is like the symmetric case, but we need to be careful:
1662 # * We want conjugate-symmetry, not just symmetry.
1663 # * The diagonal will (as a result) be real.
1667 for j
in range(i
+1):
1668 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1670 Sij
= cls
.real_embed(Eij
)
1673 # The second one has a minus because it's conjugated.
1674 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1676 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1679 # Since we embedded these, we can drop back to the "field" that we
1680 # started with instead of the complex extension "F".
1681 return ( s
.change_ring(field
) for s
in S
)
1684 def __init__(self
, n
, field
=AA
, **kwargs
):
1685 basis
= self
._denormalized
_basis
(n
,field
)
1686 super(ComplexHermitianEJA
,self
).__init
__(field
, basis
, n
, **kwargs
)
1689 class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1693 Embed the n-by-n quaternion matrix ``M`` into the space of real
1694 matrices of size 4n-by-4n by first sending each quaternion entry `z
1695 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1696 c+di],[-c + di, a-bi]]`, and then embedding those into a real
1701 sage: from mjo.eja.eja_algebra import \
1702 ....: QuaternionMatrixEuclideanJordanAlgebra
1706 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1707 sage: i,j,k = Q.gens()
1708 sage: x = 1 + 2*i + 3*j + 4*k
1709 sage: M = matrix(Q, 1, [[x]])
1710 sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1716 Embedding is a homomorphism (isomorphism, in fact)::
1718 sage: set_random_seed()
1719 sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
1720 sage: n = ZZ.random_element(n_max)
1721 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1722 sage: X = random_matrix(Q, n)
1723 sage: Y = random_matrix(Q, n)
1724 sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
1725 sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
1726 sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1731 quaternions
= M
.base_ring()
1734 raise ValueError("the matrix 'M' must be square")
1736 F
= QuadraticField(-1, 'I')
1741 t
= z
.coefficient_tuple()
1746 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1747 [-c
+ d
*i
, a
- b
*i
]])
1748 realM
= ComplexMatrixEuclideanJordanAlgebra
.real_embed(cplxM
)
1749 blocks
.append(realM
)
1751 # We should have real entries by now, so use the realest field
1752 # we've got for the return value.
1753 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1758 def real_unembed(M
):
1760 The inverse of _embed_quaternion_matrix().
1764 sage: from mjo.eja.eja_algebra import \
1765 ....: QuaternionMatrixEuclideanJordanAlgebra
1769 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1770 ....: [-2, 1, -4, 3],
1771 ....: [-3, 4, 1, -2],
1772 ....: [-4, -3, 2, 1]])
1773 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
1774 [1 + 2*i + 3*j + 4*k]
1778 Unembedding is the inverse of embedding::
1780 sage: set_random_seed()
1781 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1782 sage: M = random_matrix(Q, 3)
1783 sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1784 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1790 raise ValueError("the matrix 'M' must be square")
1791 if not n
.mod(4).is_zero():
1792 raise ValueError("the matrix 'M' must be a quaternion embedding")
1794 # Use the base ring of the matrix to ensure that its entries can be
1795 # multiplied by elements of the quaternion algebra.
1796 field
= M
.base_ring()
1797 Q
= QuaternionAlgebra(field
,-1,-1)
1800 # Go top-left to bottom-right (reading order), converting every
1801 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1804 for l
in range(n
/4):
1805 for m
in range(n
/4):
1806 submat
= ComplexMatrixEuclideanJordanAlgebra
.real_unembed(
1807 M
[4*l
:4*l
+4,4*m
:4*m
+4] )
1808 if submat
[0,0] != submat
[1,1].conjugate():
1809 raise ValueError('bad on-diagonal submatrix')
1810 if submat
[0,1] != -submat
[1,0].conjugate():
1811 raise ValueError('bad off-diagonal submatrix')
1812 z
= submat
[0,0].real()
1813 z
+= submat
[0,0].imag()*i
1814 z
+= submat
[0,1].real()*j
1815 z
+= submat
[0,1].imag()*k
1818 return matrix(Q
, n
/4, elements
)
1822 def natural_inner_product(cls
,X
,Y
):
1824 Compute a natural inner product in this algebra directly from
1829 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1833 This gives the same answer as the slow, default method implemented
1834 in :class:`MatrixEuclideanJordanAlgebra`::
1836 sage: set_random_seed()
1837 sage: J = QuaternionHermitianEJA.random_instance()
1838 sage: x,y = J.random_elements(2)
1839 sage: Xe = x.natural_representation()
1840 sage: Ye = y.natural_representation()
1841 sage: X = QuaternionHermitianEJA.real_unembed(Xe)
1842 sage: Y = QuaternionHermitianEJA.real_unembed(Ye)
1843 sage: expected = (X*Y).trace().coefficient_tuple()[0]
1844 sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye)
1845 sage: actual == expected
1849 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/4
1852 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra
,
1855 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1856 matrices, the usual symmetric Jordan product, and the
1857 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1862 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1866 In theory, our "field" can be any subfield of the reals::
1868 sage: QuaternionHermitianEJA(2, RDF)
1869 Euclidean Jordan algebra of dimension 6 over Real Double Field
1870 sage: QuaternionHermitianEJA(2, RR)
1871 Euclidean Jordan algebra of dimension 6 over Real Field with
1872 53 bits of precision
1876 The dimension of this algebra is `2*n^2 - n`::
1878 sage: set_random_seed()
1879 sage: n_max = QuaternionHermitianEJA._max_test_case_size()
1880 sage: n = ZZ.random_element(1, n_max)
1881 sage: J = QuaternionHermitianEJA(n)
1882 sage: J.dimension() == 2*(n^2) - n
1885 The Jordan multiplication is what we think it is::
1887 sage: set_random_seed()
1888 sage: J = QuaternionHermitianEJA.random_instance()
1889 sage: x,y = J.random_elements(2)
1890 sage: actual = (x*y).natural_representation()
1891 sage: X = x.natural_representation()
1892 sage: Y = y.natural_representation()
1893 sage: expected = (X*Y + Y*X)/2
1894 sage: actual == expected
1896 sage: J(expected) == x*y
1899 We can change the generator prefix::
1901 sage: QuaternionHermitianEJA(2, prefix='a').gens()
1902 (a0, a1, a2, a3, a4, a5)
1904 Our natural basis is normalized with respect to the natural inner
1905 product unless we specify otherwise::
1907 sage: set_random_seed()
1908 sage: J = QuaternionHermitianEJA.random_instance()
1909 sage: all( b.norm() == 1 for b in J.gens() )
1912 Since our natural basis is normalized with respect to the natural
1913 inner product, and since we know that this algebra is an EJA, any
1914 left-multiplication operator's matrix will be symmetric because
1915 natural->EJA basis representation is an isometry and within the EJA
1916 the operator is self-adjoint by the Jordan axiom::
1918 sage: set_random_seed()
1919 sage: x = QuaternionHermitianEJA.random_instance().random_element()
1920 sage: x.operator().matrix().is_symmetric()
1925 def _denormalized_basis(cls
, n
, field
):
1927 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1929 Why do we embed these? Basically, because all of numerical
1930 linear algebra assumes that you're working with vectors consisting
1931 of `n` entries from a field and scalars from the same field. There's
1932 no way to tell SageMath that (for example) the vectors contain
1933 complex numbers, while the scalar field is real.
1937 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1941 sage: set_random_seed()
1942 sage: n = ZZ.random_element(1,5)
1943 sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
1944 sage: all( M.is_symmetric() for M in B )
1948 Q
= QuaternionAlgebra(QQ
,-1,-1)
1951 # This is like the symmetric case, but we need to be careful:
1953 # * We want conjugate-symmetry, not just symmetry.
1954 # * The diagonal will (as a result) be real.
1958 for j
in range(i
+1):
1959 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1961 Sij
= cls
.real_embed(Eij
)
1964 # The second, third, and fourth ones have a minus
1965 # because they're conjugated.
1966 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1968 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1970 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
1972 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
1975 # Since we embedded these, we can drop back to the "field" that we
1976 # started with instead of the quaternion algebra "Q".
1977 return ( s
.change_ring(field
) for s
in S
)
1980 def __init__(self
, n
, field
=AA
, **kwargs
):
1981 basis
= self
._denormalized
_basis
(n
,field
)
1982 super(QuaternionHermitianEJA
,self
).__init
__(field
, basis
, n
, **kwargs
)
1985 class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra
, KnownRankEJA
):
1987 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1988 with the half-trace inner product and jordan product ``x*y =
1989 (x0*y0 + <B*x_bar,y_bar>, x0*y_bar + y0*x_bar)`` where ``B`` is a
1990 symmetric positive-definite "bilinear form" matrix. It has
1991 dimension `n` over the reals, and reduces to the ``JordanSpinEJA``
1992 when ``B`` is the identity matrix of order ``n-1``.
1996 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
1997 ....: JordanSpinEJA)
2001 When no bilinear form is specified, the identity matrix is used,
2002 and the resulting algebra is the Jordan spin algebra::
2004 sage: J0 = BilinearFormEJA(3)
2005 sage: J1 = JordanSpinEJA(3)
2006 sage: J0.multiplication_table() == J0.multiplication_table()
2011 We can create a zero-dimensional algebra::
2013 sage: J = BilinearFormEJA(0)
2017 We can check the multiplication condition given in the Jordan, von
2018 Neumann, and Wigner paper (and also discussed on my "On the
2019 symmetry..." paper). Note that this relies heavily on the standard
2020 choice of basis, as does anything utilizing the bilinear form matrix::
2022 sage: set_random_seed()
2023 sage: n = ZZ.random_element(5)
2024 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
2025 sage: B = M.transpose()*M
2026 sage: J = BilinearFormEJA(n, B=B)
2027 sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
2028 sage: V = J.vector_space()
2029 sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
2030 ....: for ei in eis ]
2031 sage: actual = [ sis[i]*sis[j]
2032 ....: for i in range(n-1)
2033 ....: for j in range(n-1) ]
2034 sage: expected = [ J.one() if i == j else J.zero()
2035 ....: for i in range(n-1)
2036 ....: for j in range(n-1) ]
2037 sage: actual == expected
2040 def __init__(self
, n
, field
=AA
, B
=None, **kwargs
):
2042 self
._B
= matrix
.identity(field
, max(0,n
-1))
2046 V
= VectorSpace(field
, n
)
2047 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
2056 z0
= x0
*y0
+ (self
._B
*xbar
).inner_product(ybar
)
2057 zbar
= y0
*xbar
+ x0
*ybar
2058 z
= V([z0
] + zbar
.list())
2059 mult_table
[i
][j
] = z
2061 # The rank of this algebra is two, unless we're in a
2062 # one-dimensional ambient space (because the rank is bounded
2063 # by the ambient dimension).
2064 fdeja
= super(BilinearFormEJA
, self
)
2065 return fdeja
.__init
__(field
, mult_table
, rank
=min(n
,2), **kwargs
)
2067 def inner_product(self
, x
, y
):
2069 Half of the trace inner product.
2071 This is defined so that the special case of the Jordan spin
2072 algebra gets the usual inner product.
2076 sage: from mjo.eja.eja_algebra import BilinearFormEJA
2080 Ensure that this is one-half of the trace inner-product when
2081 the algebra isn't just the reals (when ``n`` isn't one). This
2082 is in Faraut and Koranyi, and also my "On the symmetry..."
2085 sage: set_random_seed()
2086 sage: n = ZZ.random_element(2,5)
2087 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
2088 sage: B = M.transpose()*M
2089 sage: J = BilinearFormEJA(n, B=B)
2090 sage: x = J.random_element()
2091 sage: y = J.random_element()
2092 sage: x.inner_product(y) == (x*y).trace()/2
2096 xvec
= x
.to_vector()
2098 yvec
= y
.to_vector()
2100 return x
[0]*y
[0] + (self
._B
*xbar
).inner_product(ybar
)
2103 class JordanSpinEJA(BilinearFormEJA
):
2105 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2106 with the usual inner product and jordan product ``x*y =
2107 (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2112 sage: from mjo.eja.eja_algebra import JordanSpinEJA
2116 This multiplication table can be verified by hand::
2118 sage: J = JordanSpinEJA(4)
2119 sage: e0,e1,e2,e3 = J.gens()
2135 We can change the generator prefix::
2137 sage: JordanSpinEJA(2, prefix='B').gens()
2142 Ensure that we have the usual inner product on `R^n`::
2144 sage: set_random_seed()
2145 sage: J = JordanSpinEJA.random_instance()
2146 sage: x,y = J.random_elements(2)
2147 sage: X = x.natural_representation()
2148 sage: Y = y.natural_representation()
2149 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
2153 def __init__(self
, n
, field
=AA
, **kwargs
):
2154 # This is a special case of the BilinearFormEJA with the identity
2155 # matrix as its bilinear form.
2156 return super(JordanSpinEJA
, self
).__init
__(n
, field
, **kwargs
)
2159 class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra
, KnownRankEJA
):
2161 The trivial Euclidean Jordan algebra consisting of only a zero element.
2165 sage: from mjo.eja.eja_algebra import TrivialEJA
2169 sage: J = TrivialEJA()
2176 sage: 7*J.one()*12*J.one()
2178 sage: J.one().inner_product(J.one())
2180 sage: J.one().norm()
2182 sage: J.one().subalgebra_generated_by()
2183 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
2188 def __init__(self
, field
=AA
, **kwargs
):
2190 fdeja
= super(TrivialEJA
, self
)
2191 # The rank is zero using my definition, namely the dimension of the
2192 # largest subalgebra generated by any element.
2193 return fdeja
.__init
__(field
, mult_table
, rank
=0, **kwargs
)