Matrix of ones

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

$J_{2}={\begin{pmatrix}1&1\\1&1\end{pmatrix}};\quad J_{3}={\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}};\quad J_{2,5}={\begin{pmatrix}1&1&1&1&1\\1&1&1&1&1\end{pmatrix}}.\quad$

Some sources call the all-ones matrix the unit matrix, but that term may also refer to the identity matrix, a different matrix.

Properties

For an n×n matrix of ones J, the following properties hold:

• The characteristic polynomial of J is $(x-n)x^{n-1}$.
• The trace of J is n, and the determinant is 1 if n is 1, or 0 otherwise.
• The rank of J is 1 and the eigenvalues are n with multiplicity 1 and 0 with multiplicity n-1.
• J is positive semi-definite matrix. This follows from the previous property.
• $J^{k}=n^{k-1}J,{\mbox{ for }}k=1,2,\ldots .\,$
• The matrix ${\tfrac {1}{n}}J$ is idempotent. This is a simple corollary of the above.
• $\exp(J)=I+{\frac {e^{n}-1}{n}}J,$ where exp(J) is the matrix exponential.
• J is the neutral element of the Hadamard product.
• If A is the adjacency matrix of a 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.