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:

$J_2=\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix};\quad J_3=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix};\quad J_{2,5}=\begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{pmatrix}.\quad$

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 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 (once) and 0 (n-1 times).
• 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 $\tfrac1n J$ 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.