Matrix of ones

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


 * $$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, 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 $n&thinsp;×&thinsp;n$ matrix of ones J, the following properties hold:


 * The trace of J equals n, 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 $n − 1$.
 * $$ J^k = n^{k-1} J$$ for $$k = 1,2,\ldots .$$
 * J is the neutral element of the Hadamard product.

When J is considered as a matrix over the real numbers, the following additional properties hold:
 * J is positive semi-definite matrix.
 * The matrix $$\tfrac1n J$$ is idempotent.
 * The matrix exponential of J is $$\exp(\mu J)=I+\frac{e^{\mu n}-1}{n}J$$

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. 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.