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In mathematics, a matrix of ones or all-ones matrix is a matrix with every entry equal to one.^[1] For example:

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

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 $n \times n$ matrix of ones J, the following properties hold:

The trace of J equals n,^[3] 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 $n - 1$ .^[4]
$J^{k}=n^{k-1}J$ for $k=1,2,\ldots .$ ^[5]
J is the neutral element of the Hadamard product.^[6]

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

J is positive semi-definite matrix.
The matrix ${\tfrac {1}{n}}J$ is idempotent.^[5]
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.^[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.

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 $(a\cdot b)\cdot (b\cdot c)=b$ . Finite central groupoids have a square number of elements, and the corresponding logical matrices exist only for those dimensions.^[8]