mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a matrix of ones or all-ones matrix is a

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

with every entry equal to

one 1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sp ...

. For example: :

J_2 = \begin
1 & 1 \\
1 & 1 
\end,\quad
J_3 = \begin
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end,\quad
J_ = \begin
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 
\end,\quad
J_ = \begin
1 & 1 
\end.\quad

Some sources call the all-ones matrix the unit matrix, but that term may also refer to the

identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...

, 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 vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...

s''.

Properties

For an matrix of ones ''J'', the following properties hold: * The trace of ''J'' equals ''n'', and the

determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

equals 0 for ''n'' ≥ 2, but equals 1 if ''n'' = 1. * The

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

of ''J'' is

(x - n)x^

. * The minimal polynomial of ''J'' is

x^2-nx

. * The rank of ''J'' is 1 and the

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

s are ''n'' with multiplicity 1 and 0 with multiplicity . *

J^k = n^ J

for

k = 1,2,\ldots .

. * ''J'' is the

neutral element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

of the Hadamard product. When ''J'' is considered as a matrix over the

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s, the following additional properties hold: * ''J'' is positive semi-definite matrix. *The matrix

\tfrac1n J

idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

. *The

matrix exponential In mathematics, the matrix exponential is a matrix function on square matrix, square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exp ...

of ''J'' is

\exp(\mu J)=I+\fracJ

Applications

The all-ones matrix arises in the mathematical field of

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

, particularly involving the application of algebraic methods to

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

. For example, if ''A'' is the

adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph (discrete mathematics), graph. The elements of the matrix (mathematics), matrix indicate whether pairs of Vertex (graph theory), vertices ...

of an ''n''-vertex

undirected graph In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called '' vertices'' (also call ...

''G'', and ''J'' is the all-ones matrix of the same dimension, then ''G'' is a

regular graph In graph theory, a regular graph is a Graph (discrete mathematics), graph where each Vertex (graph theory), vertex has the same number of neighbors; i.e. every vertex has the same Degree (graph theory), degree or valency. A regular directed graph ...

if and only if ''AJ'' = ''JA''. As a second example, the matrix appears in some linear-algebraic proofs of

Cayley's formula In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer n, the number of trees on n labeled vertices is n^. The formula equivalently counts the spanning trees of a ...

, which gives the number of

spanning tree In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is no ...

s of a

complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices i ...

, using the

matrix tree theorem In the mathematics, mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph (discrete mathematics), graph, showing that this ...

. The logical

square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

s 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\cdot b)\cdot (b\cdot c)=b

. Finite central groupoids have a

square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...

of elements, and the corresponding logical matrices exist only for those dimensions.

References

Matrices (mathematics) 1 (number) {{Linear-algebra-stub

Properties

Applications

See also

References