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are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.
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1008:
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30:
This article is about zeros of a quadratic form. For the zero element in a vector space, see
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35:
1004:
951:
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1028:
Prolegomena to
Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions
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517:
195:
882:
804:
483:
136:
998:
943:
886:
192:
132:
31:
905:
487:
41:
871:
773:, with their null cones representing the light tracking into and out of 0 ∈
17:
409:{\displaystyle \bigcup _{r\geq 0}\{x=a+b:q(a)=-q(b)=r,\ \ a,b\in B\}.}
247:
A pseudo-Euclidean vector space may be decomposed (non-uniquely) into
40:
944:
A Singularity-free solution of the
Maxwell-Einstein Equations
455:
is a null vector then there is no multiplicative inverse for
524:, which commute so their product, when squared, yields +1:
750:{\displaystyle (1+hi)(1+hi)^{*}=(1+hi)(1-hi)=1-(hi)^{2}=0}
516:
as the foundation of this doubling construction due to
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533:
500:
302:
51:
978:. Translated by Burns, Robert G. Springer. p.
971:
749:
614:
508:
408:
120:
615:{\displaystyle (hi)^{2}=h^{2}i^{2}=(-1)(-1)=+1.}
520:(1919). In particular, these algebras have two
428:A composition algebra with a null vector is a
240:) = 0 has been used in quadratic spaces, and
8:
926:Arthur A. Sagle & Ralph E. Walde (1973)
400:
319:
244:for a quadratic space without null vectors.
928:Introduction to Lie Groups and Lie Algebras
296:consists of the union of balanced spheres:
121:{\displaystyle q(x,y,z)=x^{2}+y^{2}-z^{2}.}
34:. For null vectors in Minkowski space, see
974:Modern Geometry: Methods and Applications
735:
668:
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570:
560:
547:
532:
502:
501:
499:
478:, the split algebras arise in the series
307:
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96:
83:
50:
1000:Linear Algebra and Group Representations
27:Vector on which a quadratic form is zero
898:
416:The null cone is also the union of the
36:Minkowski space § Causal structure
950:9(2):161–8, especially 166, link from
948:Communications in Mathematical Physics
7:
874:. Null vectors are also used in the
443:, +, ×, *), the quadratic form is q(
224:which has a null vector is called a
870:for the subspace used to represent
25:
878:approach to spacetime manifolds.
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1023:Neville, E. H. (Eric Harold)
509:{\displaystyle \mathbb {C} }
757:so 1 + hi is a null vector.
476:Cayley–Dickson construction
1075:
1033:Cambridge University Press
29:
280:and negative-definite on
204:isotropic quadratic forms
889:there are null vectors.
876:Newman–Penrose formalism
276:is positive-definite on
200:definite quadratic forms
761:The real subalgebras,
751:
616:
510:
410:
226:pseudo-Euclidean space
169:is a non-zero element
128:
122:
997:Shaw, Ronald (1982).
942:Patrick Dolan (1968)
846:are null vectors and
763:split complex numbers
752:
617:
511:
411:
123:
44:
802:linearly independent
628:
531:
498:
420:through the origin.
300:
249:orthogonal subspaces
49:
437:composition algebra
142:with an associated
797:are null vectors.
779:spacetime topology
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612:
506:
406:
318:
230:isotropic vector v
129:
118:
45:A null cone where
962:Dubrovin, B. A.;
911:Geometric Algebra
767:split quaternions
490:, which uses the
480:bicomplex numbers
384:
381:
303:
242:anisotropic space
191:In the theory of
16:(Redirected from
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1059:Quadratic forms
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1007:. p. 151.
1003:. Vol. 1.
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866:can serve as a
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795:Minkowski space
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771:split-octonions
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522:imaginary units
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418:isotropic lines
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1054:Linear algebra
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1005:Academic Press
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968:Novikov, S. P.
964:Fomenko, A. T.
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952:Project Euclid
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932:Academic Press
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144:quadratic form
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518:L. E. Dickson
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883:Verma module
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473:
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459:, and since
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139:
137:vector space
130:
887:Lie algebra
793:vectors of
488:bioctonions
228:. The term
163:null vector
133:mathematics
32:Zero vector
1048:Categories
1035:. p.
906:Emil Artin
893:References
791:light-like
777:, suggest
177:for which
149:, written
135:, given a
916:isotropic
872:spacetime
800:The four
720:−
702:−
670:∗
595:−
583:−
467:is not a
395:∈
356:−
312:≥
305:⋃
286:null cone
103:−
18:Null cone
1025:(1922).
970:(1984).
785:Examples
451:*. When
272:, where
908:(1957)
881:In the
474:In the
1011:
986:
841:= 1 –
836:, and
831:= 1 +
821:= 1 +
811:= 1 +
769:, and
494:field
486:, and
383:
380:
284:. The
885:of a
868:basis
463:≠ 0,
435:In a
292:, of
288:, or
232:when
186:) = 0
1009:ISBN
984:ISBN
789:The
622:Then
447:) =
254:and
202:and
193:real
161:, a
1037:204
449:x x
173:of
165:or
131:In
1050::
1031:.
982:.
980:50
966:;
946:,
914:,
860:,
856:,
852:,
848:{
843:hk
833:hk
826:,
823:hj
816:,
813:hi
781:.
765:,
610:1.
482:,
471:.
432:.
267:+
263:=
258:,
218:,
209:A
198:,
188:.
155:,
1039:.
1017:.
992:.
864:}
862:m
858:m
854:n
850:l
839:m
829:m
819:n
809:l
775:A
745:0
742:=
737:2
733:)
729:i
726:h
723:(
717:1
714:=
711:)
708:i
705:h
699:1
696:(
693:)
690:i
687:h
684:+
681:1
678:(
675:=
666:)
662:i
659:h
656:+
653:1
650:(
647:)
644:i
641:h
638:+
635:1
632:(
607:+
604:=
601:)
598:1
592:(
589:)
586:1
580:(
577:=
572:2
568:i
562:2
558:h
554:=
549:2
545:)
541:i
538:h
535:(
503:C
465:A
461:x
457:x
453:x
445:x
441:A
439:(
404:.
401:}
398:B
392:b
389:,
386:a
377:,
374:r
371:=
368:)
365:b
362:(
359:q
353:=
350:)
347:a
344:(
341:q
338::
335:b
332:+
329:a
326:=
323:x
320:{
315:0
309:r
294:X
282:B
278:A
274:q
269:B
265:A
261:X
256:B
252:A
238:v
236:(
234:q
222:)
220:q
216:X
214:(
184:x
182:(
180:q
175:X
171:x
159:)
157:q
153:X
151:(
147:q
140:X
116:.
111:2
107:z
98:2
94:y
90:+
85:2
81:x
77:=
74:)
71:z
68:,
65:y
62:,
59:x
56:(
53:q
38:.
20:)
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