Editing Matrix of ones

{{Short description|Matrix with every entry equal to one}}
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In [[mathematics]], a '''matrix of ones''' or '''all-ones matrix''' is a [[Matrix (mathematics)|matrix]] with every entry equal to [[1 (number)|one]].<ref>{{citation|title=Matrix Analysis|first1=Roger A.|last1=Horn|first2=Charles R.|last2=Johnson|author2-link= Charles Royal Johnson |publisher=Cambridge University Press|year= 2012|isbn=9780521839402|page=8|url=https://books.google.com/books?id=5I5AYeeh0JUC&pg=PA8|contribution=0.2.8 The all-ones matrix and vector}}.</ref> For example:

:<math>J_2 = \begin{bmatrix}
1 & 1 \\
1 & 1 
\end{bmatrix},\quad
J_3 = \begin{bmatrix}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{bmatrix},\quad
J_{2,5} = \begin{bmatrix}
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 
\end{bmatrix},\quad
J_{1,2} = \begin{bmatrix}
1 & 1 
\end{bmatrix}.\quad</math>

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 type of matrix.

A '''vector of ones''' or '''all-ones vector''' is matrix of ones having [[row and column vectors|row or column form]]; it should not be confused with ''[[unit vector]]s''.

==Properties==
For an {{math|''n''&thinsp;×&thinsp;''n''}} matrix of ones ''J'', the following properties hold:

* The [[trace (linear algebra) | trace]] of ''J'' equals ''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=https://books.google.com/books?id=_Tc_AAAAQBAJ&pg=PA4|at=Lemma 1.4, p.&nbsp;4}}.</ref> and the [[determinant]] equals 0 for ''n'' ≥ 2, but equals 1 if ''n'' = 1.
* The [[characteristic polynomial]] of ''J'' is <math>(x - n)x^{n-1}</math>.
* The [[Minimal_polynomial_(linear_algebra)|minimal polynomial]] of ''J'' is <math>x^2-nx</math>.
* The [[rank of a matrix|rank]] of ''J'' is 1 and the [[eigenvalue]]s are ''n'' with [[Eigenvalues_and_eigenvectors#Algebraic_multiplicity|multiplicity]] 1 and 0 with  multiplicity {{math|''n'' − 1}}.<ref>{{harvtxt|Stanley|2013}}; {{harvtxt|Horn|Johnson|2012}}, [https://books.google.com/books?id=5I5AYeeh0JUC&pg=PA65 p.&nbsp;65].</ref>
* <math> J^k = n^{k-1} J</math> for <math>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=https://books.google.com/books?id=vtiyg6fnnskC&pg=PA30}}.</ref>
* ''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=https://books.google.com/books?id=PQUAQh04lrUC&pg=PA77}}.</ref>

When ''J'' is considered as a matrix over the [[real number]]s, the following additional properties hold:
* ''J'' is [[Definite symmetric matrix|positive semi-definite matrix]].
*The matrix <math>\tfrac1n J</math> is [[Idempotent matrix|idempotent]].<ref name="timm"/>
*The [[matrix exponential]] of ''J'' is <math>\exp(\mu J)=I+\frac{e^{\mu n}-1}{n}J</math>

==Applications==
The all-ones matrix arises in the mathematical field of [[combinatorics]], particularly involving the application of algebraic methods to [[graph theory]].  For example, if ''A'' is the [[adjacency matrix]] of an ''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''&nbsp;=&nbsp;''JA''.<ref>{{citation|title=Algebraic Combinatorics|first=Chris|last=Godsil|authorlink= Chris Godsil |publisher=CRC Press|year=1993|isbn=9780412041310|url=https://books.google.com/books?id=eADtlNCkkIMC&pg=PA25|at=Lemma 4.1, p.&nbsp;25}}.</ref>  As a second example, the matrix appears in some linear-algebraic proofs of [[Cayley's formula]], which gives the number of [[spanning tree]]s of a [[complete graph]], using the [[matrix tree theorem]].

The logical [[square root]]s of a matrix of ones, [[logical matrix|logical matrices]] whose square is a matrix of ones, can be used to characterize the [[central groupoid]]s. Central groupoids are algebraic structures that obey the [[identity (mathematics)|identity]] <math>(a\cdot b)\cdot (b\cdot c)=b</math>. Finite central groupoids have a [[square number]] of elements, and the corresponding logical matrices exist only for those dimensions.<ref>{{citation
 | last = Knuth | first = Donald E. | author-link = Donald Knuth
 | doi = 10.1016/S0021-9800(70)80032-1
 | journal = [[Journal of Combinatorial Theory]]
 | mr = 259000
 | pages = 376–390
 | title = Notes on central groupoids
 | volume = 8
 | year = 1970| issue = 4 }}</ref>

==See also==
* [[Zero matrix]], a matrix where all entries are zero
* [[Single-entry matrix]]

==References==
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{{notelist}}

{{Matrix classes}}

[[Category:Matrices (mathematics)]]
[[Category:1 (number)]]


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