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| In [[mathematics]], a '''matrix of ones''' or '''all-ones matrix''' is a [[Matrix (mathematics)|matrix]] where every element is equal to one.<ref>{{citation|title=Matrix Analysis|first1=Roger A.|last1=Horn|first2=Charles R.|last2=Johnson|publisher=Cambridge University Press|year= 2012|isbn=9780521839402|page=8|url=http://books.google.com/books?id=5I5AYeeh0JUC&pg=PA8|contribution=0.2.8 The all-ones matrix and vector}}.</ref> Examples of standard notation are given below:
| | I am Marion from Koloona. I love to play Clarinet. Other hobbies are Gymnastics.<br><br>My page; magnetic messaging reviews ([http://magneticmessagingbook.com Going in magneticmessagingbook.com]) |
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| :<math>J_2=\begin{pmatrix}
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| 1 & 1 \\
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| 1 & 1
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| \end{pmatrix};\quad
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| J_3=\begin{pmatrix}
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| 1 & 1 & 1 \\
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| 1 & 1 & 1 \\
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| 1 & 1 & 1
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| \end{pmatrix};\quad
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| J_{2,5}=\begin{pmatrix}
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| 1 & 1 & 1 & 1 & 1 \\
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| 1 & 1 & 1 & 1 & 1
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| \end{pmatrix}.\quad</math>
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| Some sources call the all-ones matrix the '''unit matrix''',<ref>{{MathWorld|title=Unit Matrix|urlname=UnitMatrix}}</ref> but that term may also refer to the [[identity matrix]], a different matrix.
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| ==Properties==
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| For an ''n×n'' matrix of ones ''J'', the following properties hold:
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| * The [[trace (linear algebra) | trace]] of ''J'' is ''n'',<ref>{{citation|title=Algebraic Combinatorics: Walks, Trees, Tableaux, and More|publisher=Springer|year=2013|isbn=9781461469988|first=Richard P.|last=Stanley|authorlink=Richard P. Stanley|url=http://books.google.com/books?id=_Tc_AAAAQBAJ&pg=PA4|at=Lemma 1.4, p. 4}}.</ref> and the [[determinant]] is 1 if ''n'' is 1, or 0 otherwise.
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| * The rank of ''J'' is 1 and the eigenvalues are ''n'' (once) and 0 (''n''-1 times).<ref>{{harvtxt|Stanley|2013}}; {{harvtxt|Horn|Johnson|2012}}, [http://books.google.com/books?id=5I5AYeeh0JUC&pg=PA65 p. 65].</ref>
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| *<math> J^k = n^{k-1} J, \mbox{ for } k=1,2,\ldots.\,</math><ref name="timm">{{citation|title=Applied Multivariate Analysis|series=Springer texts in statistics|first=Neil H.|last=Timm|publisher=Springer|year=2002|isbn=9780387227719|page=30|url=http://books.google.com/books?id=vtiyg6fnnskC&pg=PA30}}.</ref>
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| *The matrix <math>\tfrac1n J</math> is [[idempotent]]. This is a simple corollary of the above.<ref name="timm"/>
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| *<math> \exp(J) = I + \frac{ e^n-1}{n} J,</math> where exp(''J'') is the [[matrix exponential]].
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| * ''J'' is the [[neutral element]] of the [[Hadamard product (matrices)|Hadamard product]].<ref>{{citation|title=Introduction to Abstract Algebra|first=Jonathan D. H.|last=Smith|publisher=CRC Press|year=2011|isbn=9781420063721|page=77|url=http://books.google.com/books?id=PQUAQh04lrUC&pg=PA77}}.</ref>
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| *If ''A'' is the [[adjacency matrix]] of a ''n''-vertex [[undirected graph]] ''G'', and ''J'' is the all-ones matrix of the same dimension, then ''G'' is a [[regular graph]] if and only if ''AJ'' = ''JA''.<ref>{{citation|title=Algebraic Combinatorics|first=Chris|last=Godsil|publisher=CRC Press|year=1993|isbn=9780412041310|url=http://books.google.com/books?id=eADtlNCkkIMC&pg=PA25|at=Lemma 4.1, p. 25}}.</ref>
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| ==References==
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| {{reflist}}
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| [[Category:Matrices]]
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| [[Category:One]]
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| {{Linear-algebra-stub}}
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I am Marion from Koloona. I love to play Clarinet. Other hobbies are Gymnastics.
My page; magnetic messaging reviews (Going in magneticmessagingbook.com)