Matrix of ones

In mathematics, a matrix of ones or all-ones matrix is a matrix where every element is equal to one. Examples of standard notation are given below:
Some sources call the all-ones matrix the unit matrix, but that term may also refer to the identity matrix, a different matrix.
A vector of ones or all-ones vector is matrix of ones having row or column form.
Properties.
For an 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". 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.