Matrix of ones

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 (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 . • 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. 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. ==See also==