Matrix of ones

In mathematics, a matrix of ones is a matrix where every element is equal to one. Examples of standard notation are given below:

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

In special contexts, the term unit matrix is used as a synonym for "matrix of ones"^[1] This is done whenever it is clear that "unit matrix" does not refer to the identity matrix.

Properties

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

The trace of U is n, and the determinant is 1 if n is 1, or 0 otherwise.
The rank of U is 1 and the eigenvalues are n (once) and 0 (n-1 times).
$U^k = n^{k-1} U, \mbox{ for } k=1,2,\ldots.\,$
The matrix $\tfrac1n U$ is idempotent. This is a simple corollary of the above.
$\operatorname{exp}(U) = I %2B \frac{ e^n-1}{n} U,$ where exp(U) is the matrix exponential.
Multiplication by U with the Hadamard product is the identity operator.

References

^ Weisstein, Eric W., "Unit Matrix" from MathWorld.