matrix of ones

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

• The trace of J equals n,{{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. 4}}. and the determinant equals 0 for n ≥ 2, but equals 1 if n = 1.
• The characteristic polynomial of J is $(x\; -\; n)x^\{n-1\}$.
• The minimal polynomial of J is $x^2-nx$.
• The rank of J is 1 and the eigenvalues are n with multiplicity 1 and 0 with multiplicity {{math|n − 1}}.{{harvtxt|Stanley|2013}}; {{harvtxt|Horn|Johnson|2012}}, [https://books.google.com/books?id=5I5AYeeh0JUC&pg=PA65 p. 65].
• $J^k\; =\; n^\{k-1\}\; J$ for $k\; =\; 1,2,\backslash ldots\; .${{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}}.
• J is the neutral element of the Hadamard product.{{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}}.

When J is considered as a matrix over the real numbers, the following additional properties hold:

• J is positive semi-definite matrix.
• The matrix $\backslash tfrac1n\; J$ is idempotent.
• The matrix exponential of J is $\backslash exp(\backslash mu\; J)=I+\backslash frac\{e^\{\backslash mu\; n\}-1\}\{n\}J$

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 = JA.{{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. 25}}. 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 $(a\backslash cdot\; b)\backslash cdot\; (b\backslash cdot\; c)=b$. Finite central groupoids have a square number of elements, and the corresponding logical matrices exist only for those dimensions.{{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 }}

• Zero matrix, a matrix where all entries are zero
• Single-entry matrix

{{reflist}}

{{notelist}}

{{Matrix classes}}

Category:Matrices (mathematics)

Category:1 (number)

{{Linear-algebra-stub}}