1917:
43:
1493:
1106:
1676:, the zero matrix serves the role of both an additive identity and an absorbing element. In general, the zero element of a ring is unique, and typically denoted as 0 without any subscript to indicate the parent ring. Hence the examples above represent zero matrices over any ring.
1296:
918:
1488:{\displaystyle 0_{K_{m,n}}={\begin{bmatrix}0_{K}&0_{K}&\cdots &0_{K}\\0_{K}&0_{K}&\cdots &0_{K}\\\vdots &\vdots &&\vdots \\0_{K}&0_{K}&\cdots &0_{K}\end{bmatrix}}}
1655:
1101:{\displaystyle 0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},\ 0_{2,3}={\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}},\ }
1714:
of any tensor with any zero tensor results in another zero tensor. Among tensors of a given type, the zero tensor of that type serves as the additive identity among those tensors.
521:). For example, the trivial structure (containing only the identity) is a zero object in categories where morphisms must map identities to identities. Specific examples include:
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Many absorbing elements are also additive identities, including the empty set and the zero function. Another important example is the distinguished element 0 in a
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708:, for example, zero morphisms are morphisms which always return group identities, thus generalising the function
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incorrectly led you here, you may wish to change the link to point directly to the intended article.
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780:. That the zero module is in fact a module is simple to show; it is closed under addition and
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155:. These alternate meanings may or may not reduce to the same thing, depending on the context.
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569:: any morphism composed with a zero morphism gives a zero morphism. Specifically, if
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element). The fact that this is an ideal follows directly from the definition.
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includes a list of related items that share the same name (or similar names).
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may sometimes be called a zero element, and written either as 0 or ⊥.
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1707:. The zero tensor of order 1 is sometimes known as the zero vector.
181:. It corresponds to the element 0 such that for all x in the group,
542:, containing only the identity (a zero object in the category of
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1660:There is exactly one zero matrix of any given size
67:. Unsourced material may be challenged and removed.
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1922:Index of articles associated with the same name
359:(an empty coproduct, and so an identity under
212:its norm (length) is also 0. Often denoted as
208:: the vector whose components are all 0; in a
196:. Some examples of additive identity include:
1650:{\displaystyle 0_{K_{m,n}}+A=A+0_{K_{m,n}}=A}
852:consisting of only the additive identity (or
8:
1703:, of any order, all of whose components are
1683:which sends all vectors to the zero vector.
1498:The zero matrix is the additive identity in
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27:Generalizations of 0 in algebraic structures
1764:Nair, M. Thamban; Singh, Arindama (2018).
892:. It is alternately denoted by the symbol
586:is the zero morphism among morphisms from
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687:and composing them gives a zero morphism
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127:Learn how and when to remove this message
1232:is the matrix with all entries equal to
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404:, which is an absorbing element under
912:. Some examples of zero matrices are
666:, then there are canonical morphisms
147:is one of several generalizations of
7:
1679:The zero matrix also represents the
65:adding citations to reliable sources
1810:Undergraduate Texts in Mathematics
25:
1915:
760:consisting of only the additive
662:If a category has a zero object
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1824:We have a zero matrix in which
513:(and so an identity under both
52:needs additional citations for
622:are arbitrary morphisms, then
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30:For other uses of "Zero", see
1:
1563:{\displaystyle A\in K_{m,n}}
1286:is the additive identity in
227:{\displaystyle \mathbf {0} }
1192:{\displaystyle 0_{K_{m,n}}}
1119:matrices with entries in a
888:with all its entries being
511:initial and terminal object
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256:{\displaystyle {\vec {0}}}
29:
1776:10.1007/978-981-13-0926-7
386:generalises the property
1886:. ... We shall write it
1853:{\displaystyle a_{ij}=0}
1812:. Springer. p. 25.
439:pointwise multiplication
1770:. Springer. p. 3.
1746:— non-mathematical uses
1524:{\displaystyle K_{m,n}}
1225:{\displaystyle K_{m,n}}
1152:{\displaystyle K_{m,n}}
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61:improve this article
1879:{\displaystyle i,j}
1531:. That is, for all
1159:. The zero matrix
772:, this identity is
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159:Additive identities
1955:Set index articles
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1672:zero matrix. In a
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1899:{\displaystyle O}
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905:{\displaystyle O}
816:{\displaystyle R}
768:function. In the
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864:Main article:
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551:Zero morphisms
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72:Find sources:
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50:This article
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546:over a ring)
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280:
276:
271:
267:
201:
192:
188:
184:
164:
162:
145:zero element
144:
138:
123:
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
1800:Lang, Serge
1734:Zero object
1697:zero tensor
1693:mathematics
1687:Zero tensor
1674:matrix ring
1111:The set of
881:zero matrix
872:mathematics
866:Zero matrix
860:Zero matrix
794:mathematics
784:trivially.
778:zero module
754:zero module
750:mathematics
744:Zero module
540:zero module
509:is both an
502:zero object
426:defined by
274:defined by
202:zero vector
141:mathematics
117:August 2020
18:Zero tensor
1950:0 (number)
1944:Categories
1751:References
798:zero ideal
788:Zero ideal
515:coproducts
361:coproducts
87:newspapers
1710:Taking a
1542:∈
1463:⋯
1432:⋮
1426:⋮
1421:⋮
1402:⋯
1359:⋯
704:. In the
401:empty set
380:semigroup
344:coproduct
337:empty sum
330:set union
325:empty set
248:→
151:to other
1860:for all
1802:(1987).
1718:See also
1259:, where
770:integers
766:addition
762:identity
695: :
613: :
599: :
577: :
563:category
519:products
507:category
424:zero map
384:semiring
357:category
285:, under
272:zero map
823:is the
756:is the
738:lattice
544:modules
169:is the
101:scholar
1931:If an
1816:
1782:
1701:tensor
1695:, the
1096:
1025:
964:
886:matrix
796:, the
758:module
752:, the
717:) = 0.
594:, and
437:under
410:{ } ×
342:empty
328:under
204:under
191:+ 0 =
179:monoid
173:in an
103:
96:
89:
82:
74:
1924:This
1699:is a
884:is a
825:ideal
800:in a
732:in a
561:in a
505:in a
478:field
435:) = 0
414:= { }
355:in a
283:) = 0
108:JSTOR
94:books
1814:ISBN
1780:ISBN
1744:Zero
1705:zero
1290:.
1121:ring
890:zero
878:, a
854:zero
802:ring
774:zero
676:and
640:and
608:and
538:The
525:The
517:and
484:ring
462:) ⋅
454:) =
418:The
398:The
388:0 ⋅
322:The
310:) +
302:) =
266:The
200:The
183:0 +
143:, a
80:news
1772:doi
1691:In
1670:the
1199:in
870:In
792:In
748:In
736:or
652:= 0
633:= 0
627:∘ 0
590:to
481:or
422:or
392:= 0
382:or
372:An
349:An
340:or
334:An
270:or
234:or
177:or
163:An
139:In
63:by
1946::
1822:.
1808:.
1778:.
1570::
726:A
699:→
692:XY
681:→
671:→
659:.
655:AY
648:∘
645:XY
636:XB
630:XY
617:→
603:→
581:→
574:XY
555:A
499:A
450:)(
446:⋅
298:)(
294:+
187:=
1906:.
1894:O
1874:j
1871:,
1868:i
1848:0
1845:=
1840:j
1837:i
1833:a
1788:.
1774::
1666:n
1662:m
1645:A
1642:=
1635:n
1632:,
1629:m
1625:K
1620:0
1616:+
1613:A
1610:=
1607:A
1604:+
1597:n
1594:,
1591:m
1587:K
1582:0
1556:n
1553:,
1550:m
1546:K
1539:A
1517:n
1514:,
1511:m
1507:K
1481:]
1473:K
1469:0
1456:K
1452:0
1444:K
1440:0
1412:K
1408:0
1395:K
1391:0
1383:K
1379:0
1369:K
1365:0
1352:K
1348:0
1340:K
1336:0
1329:[
1324:=
1317:n
1314:,
1311:m
1307:K
1302:0
1288:K
1272:K
1268:0
1245:K
1241:0
1218:n
1215:,
1212:m
1208:K
1183:n
1180:,
1177:m
1173:K
1168:0
1145:n
1142:,
1139:m
1135:K
1124:K
1117:n
1113:m
1093:,
1088:]
1082:0
1077:0
1072:0
1065:0
1060:0
1055:0
1049:[
1044:=
1039:3
1036:,
1033:2
1029:0
1022:,
1017:]
1011:0
1006:0
999:0
994:0
988:[
983:=
978:2
975:,
972:2
968:0
961:,
956:]
950:0
944:[
939:=
934:1
931:,
928:1
924:0
900:O
840:}
837:0
834:{
811:R
715:x
713:(
711:z
701:Y
697:X
689:0
685:,
683:Y
679:0
673:0
669:X
664:0
650:f
642:0
625:g
619:B
615:Y
611:g
605:X
601:A
597:f
592:Y
588:X
583:Y
579:X
571:0
535:)
470:)
468:x
466:(
464:g
460:x
458:(
456:f
452:x
448:g
444:f
442:(
433:x
431:(
429:z
412:S
390:x
363:)
318:)
316:x
314:(
312:g
308:x
306:(
304:f
300:x
296:g
292:f
290:(
281:x
279:(
277:z
263:.
245:0
221:0
193:x
189:x
185:x
130:)
124:(
119:)
115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
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