# Matrix of ones

Short description: Matrix where every entry is equal to one

In mathematics, a matrix of ones or all-ones matrix is a matrix where every entry is equal to one.[1] Examples of standard notation are given below:

$\displaystyle{ J_2 = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix};\quad J_3 = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix};\quad J_{2,5} = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{pmatrix};\quad J_{1,2} = \begin{pmatrix} 1 & 1 \end{pmatrix}.\quad }$

Some sources call the all-ones matrix the unit matrix,[2] 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 n × n matrix of ones J, the following properties hold:

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

## 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 = JA.[7] 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.