1 from sage
.misc
.cachefunc
import cached_method
2 from sage
.combinat
.free_module
import CombinatorialFreeModule
3 from sage
.modules
.with_basis
.indexed_element
import IndexedFreeModuleElement
4 from sage
.rings
.all
import AA
6 from mjo
.matrix_algebra
import MatrixAlgebra
, MatrixAlgebraElement
8 class Octonion(IndexedFreeModuleElement
):
13 sage: from mjo.hurwitz import Octonions
18 sage: x = sum(O.gens())
20 e0 - e1 - e2 - e3 - e4 - e5 - e6 - e7
24 Conjugating twice gets you the original element::
27 sage: x = O.random_element()
28 sage: x.conjugate().conjugate() == x
32 from sage
.rings
.all
import ZZ
33 from sage
.matrix
.matrix_space
import MatrixSpace
34 C
= MatrixSpace(ZZ
,8).diagonal_matrix((1,-1,-1,-1,-1,-1,-1,-1))
35 return self
.parent().from_vector(C
*self
.to_vector())
39 Return the real part of this octonion.
41 The real part of an octonion is its projection onto the span
42 of the first generator. In other words, the "first dimension"
43 is real and the others are imaginary.
47 sage: from mjo.hurwitz import Octonions
52 sage: x = sum(O.gens())
58 This method is idempotent::
61 sage: x = O.random_element()
62 sage: x.real().real() == x.real()
66 return (self
+ self
.conjugate())/2
70 Return the imaginary part of this octonion.
72 The imaginary part of an octonion is its projection onto the
73 orthogonal complement of the span of the first generator. In
74 other words, the "first dimension" is real and the others are
79 sage: from mjo.hurwitz import Octonions
84 sage: x = sum(O.gens())
86 e1 + e2 + e3 + e4 + e5 + e6 + e7
90 This method is idempotent::
93 sage: x = O.random_element()
94 sage: x.imag().imag() == x.imag()
98 return (self
- self
.conjugate())/2
100 def _norm_squared(self
):
101 return (self
*self
.conjugate()).coefficient(0)
105 Return the norm of this octonion.
109 sage: from mjo.hurwitz import Octonions
113 sage: O = Octonions()
119 The norm is nonnegative and belongs to the base field::
121 sage: O = Octonions()
122 sage: n = O.random_element().norm()
123 sage: n >= 0 and n in O.base_ring()
126 The norm is homogeneous::
128 sage: O = Octonions()
129 sage: x = O.random_element()
130 sage: alpha = O.base_ring().random_element()
131 sage: (alpha*x).norm() == alpha.abs()*x.norm()
135 return self
._norm
_squared
().sqrt()
137 # The absolute value notation is typically used for complex numbers...
138 # and norm() isn't supported in AA, so this lets us use abs() in all
139 # of the division algebras we need.
144 Return the inverse of this element if it exists.
148 sage: from mjo.hurwitz import Octonions
152 sage: O = Octonions()
153 sage: x = sum(O.gens())
154 sage: x*x.inverse() == O.one()
159 sage: O = Octonions()
160 sage: O.one().inverse() == O.one()
165 sage: O = Octonions()
166 sage: x = O.random_element()
167 sage: x.is_zero() or ( x*x.inverse() == O.one() )
172 raise ValueError("zero is not invertible")
173 return self
.conjugate()/self
._norm
_squared
()
177 class Octonions(CombinatorialFreeModule
):
181 sage: from mjo.hurwitz import Octonions
186 Octonion algebra with base ring Algebraic Real Field
187 sage: Octonions(field=QQ)
188 Octonion algebra with base ring Rational Field
195 # Not associative, not commutative
196 from sage
.categories
.magmatic_algebras
import MagmaticAlgebras
197 category
= MagmaticAlgebras(field
).FiniteDimensional()
198 category
= category
.WithBasis().Unital()
200 super().__init
__(field
,
202 element_class
=Octonion
,
207 # The product of each basis element is plus/minus another
208 # basis element that can simply be looked up on
209 # https://en.wikipedia.org/wiki/Octonion
210 e0
, e1
, e2
, e3
, e4
, e5
, e6
, e7
= self
.gens()
211 self
._multiplication
_table
= (
212 (e0
, e1
, e2
, e3
, e4
, e5
, e6
, e7
),
213 (e1
,-e0
, e3
,-e2
, e5
,-e4
,-e7
, e6
),
214 (e2
,-e3
,-e0
, e1
, e6
, e7
,-e4
,-e5
),
215 (e3
, e2
,-e1
,-e0
, e7
,-e6
, e5
,-e4
),
216 (e4
,-e5
,-e6
,-e7
,-e0
, e1
, e2
, e3
),
217 (e5
, e4
,-e7
, e6
,-e1
,-e0
,-e3
, e2
),
218 (e6
, e7
, e4
,-e5
,-e2
, e3
,-e0
,-e1
),
219 (e7
,-e6
, e5
, e4
,-e3
,-e2
, e1
,-e0
),
222 def product_on_basis(self
, i
, j
):
223 return self
._multiplication
_table
[i
][j
]
227 Return the monomial index (basis element) corresponding to the
228 octonion unit element.
232 sage: from mjo.hurwitz import Octonions
236 This gives the correct unit element::
238 sage: O = Octonions()
239 sage: x = O.random_element()
240 sage: x*O.one() == x and O.one()*x == x
247 return ("Octonion algebra with base ring %s" % self
.base_ring())
249 def multiplication_table(self
):
251 Return a visual representation of this algebra's multiplication
252 table (on basis elements).
256 sage: from mjo.hurwitz import Octonions
260 The multiplication table is what Wikipedia says it is::
262 sage: Octonions().multiplication_table()
263 +----++----+-----+-----+-----+-----+-----+-----+-----+
264 | * || e0 | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
265 +====++====+=====+=====+=====+=====+=====+=====+=====+
266 | e0 || e0 | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
267 +----++----+-----+-----+-----+-----+-----+-----+-----+
268 | e1 || e1 | -e0 | e3 | -e2 | e5 | -e4 | -e7 | e6 |
269 +----++----+-----+-----+-----+-----+-----+-----+-----+
270 | e2 || e2 | -e3 | -e0 | e1 | e6 | e7 | -e4 | -e5 |
271 +----++----+-----+-----+-----+-----+-----+-----+-----+
272 | e3 || e3 | e2 | -e1 | -e0 | e7 | -e6 | e5 | -e4 |
273 +----++----+-----+-----+-----+-----+-----+-----+-----+
274 | e4 || e4 | -e5 | -e6 | -e7 | -e0 | e1 | e2 | e3 |
275 +----++----+-----+-----+-----+-----+-----+-----+-----+
276 | e5 || e5 | e4 | -e7 | e6 | -e1 | -e0 | -e3 | e2 |
277 +----++----+-----+-----+-----+-----+-----+-----+-----+
278 | e6 || e6 | e7 | e4 | -e5 | -e2 | e3 | -e0 | -e1 |
279 +----++----+-----+-----+-----+-----+-----+-----+-----+
280 | e7 || e7 | -e6 | e5 | e4 | -e3 | -e2 | e1 | -e0 |
281 +----++----+-----+-----+-----+-----+-----+-----+-----+
285 # Prepend the header row.
286 M
= [["*"] + list(self
.gens())]
288 # And to each subsequent row, prepend an entry that belongs to
289 # the left-side "header column."
290 M
+= [ [self
.monomial(i
)] + [ self
.monomial(i
)*self
.monomial(j
)
294 from sage
.misc
.table
import table
295 return table(M
, header_row
=True, header_column
=True, frame
=True)
301 class HurwitzMatrixAlgebraElement(MatrixAlgebraElement
):
304 Return the entrywise conjugate of this matrix.
308 sage: from mjo.hurwitz import ComplexMatrixAlgebra
312 sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
313 sage: M = A([ [ I, 1 + 2*I],
323 entries
= [ [ self
[i
,j
].conjugate()
324 for j
in range(self
.ncols())]
325 for i
in range(self
.nrows()) ]
326 return self
.parent()._element
_constructor
_(entries
)
328 def conjugate_transpose(self
):
330 Return the conjugate-transpose of this matrix.
334 sage: from mjo.hurwitz import ComplexMatrixAlgebra
338 sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
339 sage: M = A([ [ I, 2*I],
341 sage: M.conjugate_transpose()
347 sage: M.conjugate_transpose().to_vector()
348 (0, -1, 0, -3, 0, -2, 0, -4)
351 entries
= [ [ self
[j
,i
].conjugate()
352 for j
in range(self
.ncols())]
353 for i
in range(self
.nrows()) ]
354 return self
.parent()._element
_constructor
_(entries
)
356 def is_hermitian(self
):
361 sage: from mjo.hurwitz import (ComplexMatrixAlgebra,
362 ....: HurwitzMatrixAlgebra)
366 sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
367 sage: M = A([ [ 0,I],
369 sage: M.is_hermitian()
374 sage: A = HurwitzMatrixAlgebra(2, AA, QQ)
375 sage: M = A([ [1, 1],
377 sage: M.is_hermitian()
381 # A tiny bit faster than checking equality with the conjugate
383 return all( self
[i
,j
] == self
[j
,i
].conjugate()
384 for i
in range(self
.nrows())
385 for j
in range(self
.ncols()) )
388 def is_skew_hermitian(self
):
393 sage: from mjo.hurwitz import (ComplexMatrixAlgebra,
394 ....: HurwitzMatrixAlgebra)
398 sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
399 sage: M = A([ [ 0,I],
401 sage: M.is_skew_hermitian()
406 sage: A = HurwitzMatrixAlgebra(2, AA, QQ)
407 sage: M = A([ [1, 1],
409 sage: M.is_skew_hermitian()
414 sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
415 sage: M = A([ [2*I , 1 + I],
416 ....: [-1 + I, -2*I] ])
417 sage: M.is_skew_hermitian()
421 # A tiny bit faster than checking equality with the conjugate
423 return all( self
[i
,j
] == -self
[j
,i
].conjugate()
424 for i
in range(self
.nrows())
425 for j
in range(self
.ncols()) )
428 class HurwitzMatrixAlgebra(MatrixAlgebra
):
430 A class of matrix algebras whose entries come from a Hurwitz
433 For our purposes, we consider "a Hurwitz" algebra to be the real
434 or complex numbers, the quaternions, or the octonions. These are
435 typically also referred to as the Euclidean Hurwitz algebras, or
436 the normed division algebras.
438 By the Cayley-Dickson construction, each Hurwitz algebra is an
439 algebra over the real numbers, so we restrict the scalar field in
440 this case to be real. This also allows us to more accurately
441 produce the generators of the matrix algebra.
443 Element
= HurwitzMatrixAlgebraElement
445 def __init__(self
, n
, entry_algebra
, scalars
, **kwargs
):
446 from sage
.rings
.all
import RR
447 if not scalars
.is_subring(RR
):
448 # Not perfect, but it's what we're using.
449 raise ValueError("scalar field is not real")
451 super().__init
__(n
, entry_algebra
, scalars
, **kwargs
)
453 def entry_algebra_gens(self
):
455 Return a tuple of the generators of (that is, a basis for) the
456 entries of this matrix algebra.
458 This works around the inconsistency in the ``gens()`` methods
459 of the real/complex numbers, quaternions, and octonions.
463 sage: from mjo.hurwitz import Octonions, HurwitzMatrixAlgebra
467 The inclusion of the unit element is inconsistent across
468 (subalgebras of) Hurwitz algebras::
474 sage: QuaternionAlgebra(AA,1,-1).gens()
476 sage: Octonions().gens()
477 (e0, e1, e2, e3, e4, e5, e6, e7)
479 The unit element is always returned by this method, so the
480 sets of generators have cartinality 1,2,4, and 8 as you'd
483 sage: HurwitzMatrixAlgebra(2, AA, AA).entry_algebra_gens()
485 sage: HurwitzMatrixAlgebra(2, QQbar, AA).entry_algebra_gens()
487 sage: Q = QuaternionAlgebra(AA,-1,-1)
488 sage: HurwitzMatrixAlgebra(2, Q, AA).entry_algebra_gens()
490 sage: O = Octonions()
491 sage: HurwitzMatrixAlgebra(2, O, AA).entry_algebra_gens()
492 (e0, e1, e2, e3, e4, e5, e6, e7)
495 gs
= self
.entry_algebra().gens()
496 one
= self
.entry_algebra().one()
500 return (one
,) + tuple(gs
)
504 class OctonionMatrixAlgebra(HurwitzMatrixAlgebra
):
506 The algebra of ``n``-by-``n`` matrices with octonion entries over
507 (a subfield of) the real numbers.
509 The usual matrix spaces in SageMath don't support octonion entries
510 because they assume that the entries of the matrix come from a
511 commutative and associative ring, and the octonions are neither.
515 sage: from mjo.hurwitz import Octonions, OctonionMatrixAlgebra
519 sage: OctonionMatrixAlgebra(3)
520 Module of 3 by 3 matrices with entries in Octonion algebra with base
521 ring Algebraic Real Field over the scalar ring Algebraic Real Field
525 sage: OctonionMatrixAlgebra(3,scalars=QQ)
526 Module of 3 by 3 matrices with entries in Octonion algebra with
527 base ring Rational Field over the scalar ring Rational Field
531 sage: O = Octonions(RR)
532 sage: A = OctonionMatrixAlgebra(1,O)
534 Module of 1 by 1 matrices with entries in Octonion algebra with
535 base ring Real Field with 53 bits of precision over the scalar
536 ring Algebraic Real Field
538 +---------------------+
539 | 1.00000000000000*e0 |
540 +---------------------+
542 (+---------------------+
543 | 1.00000000000000*e0 |
544 +---------------------+,
545 +---------------------+
546 | 1.00000000000000*e1 |
547 +---------------------+,
548 +---------------------+
549 | 1.00000000000000*e2 |
550 +---------------------+,
551 +---------------------+
552 | 1.00000000000000*e3 |
553 +---------------------+,
554 +---------------------+
555 | 1.00000000000000*e4 |
556 +---------------------+,
557 +---------------------+
558 | 1.00000000000000*e5 |
559 +---------------------+,
560 +---------------------+
561 | 1.00000000000000*e6 |
562 +---------------------+,
563 +---------------------+
564 | 1.00000000000000*e7 |
565 +---------------------+)
569 sage: A = OctonionMatrixAlgebra(2)
570 sage: e0,e1,e2,e3,e4,e5,e6,e7 = A.entry_algebra().gens()
571 sage: A([ [e0+e4, e1+e5],
572 ....: [e2-e6, e3-e7] ])
573 +---------+---------+
574 | e0 + e4 | e1 + e5 |
575 +---------+---------+
576 | e2 - e6 | e3 - e7 |
577 +---------+---------+
581 sage: A1 = OctonionMatrixAlgebra(1,scalars=QQ)
582 sage: A2 = OctonionMatrixAlgebra(1,scalars=QQ)
583 sage: cartesian_product([A1,A2])
584 Module of 1 by 1 matrices with entries in Octonion algebra with
585 base ring Rational Field over the scalar ring Rational Field (+)
586 Module of 1 by 1 matrices with entries in Octonion algebra with
587 base ring Rational Field over the scalar ring Rational Field
591 sage: A = OctonionMatrixAlgebra(ZZ.random_element(10))
592 sage: x = A.random_element()
593 sage: x*A.one() == x and A.one()*x == x
597 def __init__(self
, n
, entry_algebra
=None, scalars
=AA
, **kwargs
):
598 if entry_algebra
is None:
599 entry_algebra
= Octonions(field
=scalars
)
605 class QuaternionMatrixAlgebra(HurwitzMatrixAlgebra
):
607 The algebra of ``n``-by-``n`` matrices with quaternion entries over
608 (a subfield of) the real numbers.
610 The usual matrix spaces in SageMath don't support quaternion entries
611 because they assume that the entries of the matrix come from a
612 commutative ring, and the quaternions are not commutative.
616 sage: from mjo.hurwitz import QuaternionMatrixAlgebra
620 sage: QuaternionMatrixAlgebra(3)
621 Module of 3 by 3 matrices with entries in Quaternion
622 Algebra (-1, -1) with base ring Algebraic Real Field
623 over the scalar ring Algebraic Real Field
627 sage: QuaternionMatrixAlgebra(3,scalars=QQ)
628 Module of 3 by 3 matrices with entries in Quaternion
629 Algebra (-1, -1) with base ring Rational Field over
630 the scalar ring Rational Field
634 sage: Q = QuaternionAlgebra(RDF, -1, -1)
635 sage: A = QuaternionMatrixAlgebra(1,Q)
637 Module of 1 by 1 matrices with entries in Quaternion Algebra
638 (-1.0, -1.0) with base ring Real Double Field over the scalar
639 ring Algebraic Real Field
660 sage: A = QuaternionMatrixAlgebra(2)
661 sage: i,j,k = A.entry_algebra().gens()
662 sage: A([ [1+i, j-2],
672 sage: A1 = QuaternionMatrixAlgebra(1,scalars=QQ)
673 sage: A2 = QuaternionMatrixAlgebra(2,scalars=QQ)
674 sage: cartesian_product([A1,A2])
675 Module of 1 by 1 matrices with entries in Quaternion Algebra
676 (-1, -1) with base ring Rational Field over the scalar ring
677 Rational Field (+) Module of 2 by 2 matrices with entries in
678 Quaternion Algebra (-1, -1) with base ring Rational Field over
679 the scalar ring Rational Field
683 sage: A = QuaternionMatrixAlgebra(ZZ.random_element(10))
684 sage: x = A.random_element()
685 sage: x*A.one() == x and A.one()*x == x
689 def __init__(self
, n
, entry_algebra
=None, scalars
=AA
, **kwargs
):
690 if entry_algebra
is None:
691 # The -1,-1 gives us the "usual" definition of quaternion
692 from sage
.algebras
.quatalg
.quaternion_algebra
import (
695 entry_algebra
= QuaternionAlgebra(scalars
,-1,-1)
696 super().__init
__(n
, entry_algebra
, scalars
, **kwargs
)
698 def _entry_algebra_element_to_vector(self
, entry
):
703 sage: from mjo.hurwitz import QuaternionMatrixAlgebra
707 sage: A = QuaternionMatrixAlgebra(2)
708 sage: u = A.entry_algebra().one()
709 sage: A._entry_algebra_element_to_vector(u)
711 sage: i,j,k = A.entry_algebra().gens()
712 sage: A._entry_algebra_element_to_vector(i)
714 sage: A._entry_algebra_element_to_vector(j)
716 sage: A._entry_algebra_element_to_vector(k)
720 from sage
.modules
.free_module
import FreeModule
721 d
= len(self
.entry_algebra_gens())
722 V
= FreeModule(self
.entry_algebra().base_ring(), d
)
723 return V(entry
.coefficient_tuple())
725 class ComplexMatrixAlgebra(HurwitzMatrixAlgebra
):
727 The algebra of ``n``-by-``n`` matrices with complex entries over
728 (a subfield of) the real numbers.
730 These differ from the usual complex matrix spaces in SageMath
731 because the scalar field is real (and not assumed to be the same
732 as the space from which the entries are drawn). The space of
733 `1`-by-`1` complex matrices will have dimension two, for example.
737 sage: from mjo.hurwitz import ComplexMatrixAlgebra
741 sage: ComplexMatrixAlgebra(3)
742 Module of 3 by 3 matrices with entries in Algebraic Field
743 over the scalar ring Algebraic Real Field
747 sage: ComplexMatrixAlgebra(3,scalars=QQ)
748 Module of 3 by 3 matrices with entries in Algebraic Field
749 over the scalar ring Rational Field
753 sage: A = ComplexMatrixAlgebra(1,CC)
755 Module of 1 by 1 matrices with entries in Complex Field with
756 53 bits of precision over the scalar ring Algebraic Real Field
762 (+------------------+
764 +------------------+,
765 +--------------------+
766 | 1.00000000000000*I |
767 +--------------------+)
771 sage: A = ComplexMatrixAlgebra(2)
772 sage: (I,) = A.entry_algebra().gens()
783 sage: A1 = ComplexMatrixAlgebra(1,scalars=QQ)
784 sage: A2 = ComplexMatrixAlgebra(2,scalars=QQ)
785 sage: cartesian_product([A1,A2])
786 Module of 1 by 1 matrices with entries in Algebraic Field over
787 the scalar ring Rational Field (+) Module of 2 by 2 matrices with
788 entries in Algebraic Field over the scalar ring Rational Field
792 sage: A = ComplexMatrixAlgebra(ZZ.random_element(10))
793 sage: x = A.random_element()
794 sage: x*A.one() == x and A.one()*x == x
798 def __init__(self
, n
, entry_algebra
=None, scalars
=AA
, **kwargs
):
799 if entry_algebra
is None:
800 from sage
.rings
.all
import QQbar
801 entry_algebra
= QQbar
802 super().__init
__(n
, entry_algebra
, scalars
, **kwargs
)
804 def _entry_algebra_element_to_vector(self
, entry
):
809 sage: from mjo.hurwitz import ComplexMatrixAlgebra
813 sage: A = ComplexMatrixAlgebra(2, QQbar, QQ)
814 sage: A._entry_algebra_element_to_vector(QQbar(1))
816 sage: A._entry_algebra_element_to_vector(QQbar(I))
820 from sage
.modules
.free_module
import FreeModule
821 d
= len(self
.entry_algebra_gens())
822 V
= FreeModule(self
.entry_algebra().base_ring(), d
)
823 return V((entry
.real(), entry
.imag()))