{{Short description|Matrix with every entry equal to one}} {{CS1 config|mode=cs2}}

In mathematics, a '''matrix of ones''' or '''all-ones matrix''' is a matrix with every entry equal to 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 or column form; it should not be confused with ''unit vectors''.

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

* The 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 of ''J'' is <math>x^2-nx</math>. * The rank of ''J'' is 1 and the eigenvalues are ''n'' with 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.<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 numbers, the following additional properties hold: * ''J'' is positive semi-definite matrix. *The matrix <math>\tfrac1n J</math> is 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 trees of a complete graph, using the matrix tree theorem.

The logical square roots of a matrix of ones, logical matrices whose square is a matrix of ones, can be used to characterize the central groupoids. Central groupoids are algebraic structures that obey the 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== {{reflist}} {{notelist}}

{{Matrix classes}}

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