Matrix of ones

In mathematics, a matrix of ones or all-ones matrix is a matrix with every entry equal to one. For example:
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 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.
The minimal polynomial of J is.
The rank of J is 1 and the eigenvalues are n with multiplicity 1 and 0 with multiplicity.
for 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 is idempotent.
The matrix exponential of J is

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.
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. Finite central groupoids have a square number of elements, and the corresponding logical matrices exist only for those dimensions.