Does this matrix shape have a name? I'm using a lot of matrices that look like the following
$$ A_3 =
\begin{bmatrix}
a & b & b\\
b & a & b\\
b & b & a
\end{bmatrix} $$
i.e., the diagonal entries are all the same, and all off-diagonal entries are the same. They need to be non-singular, so $a \ne b$, for one thing.
If you know of a common name for this shape of matrix, I would very much appreciate your help, and references would be most appreciated!
 A: Notice that a matrix with constant entries $b$ can be viewed as $b$ times the projector onto the vector $v=(1,\ldots,1)$. Thus your matrix is
\begin{align}
    A_n=(a-b)\mathbb{I}_n+b \,v v^\top.
\end{align}
Thus its eigenvalues are $a$, and $a-b$, and eigenvectors are $v$ and all vectors orthogonal to $v$.
Incidentally, $A_n$ commutes with all permutation matrices of the $n$-element permutation group in the $n$ dimensional representation. Thus it is a Casimir of that representation.
A: As far as I am aware, these matrices are called Bose Mesner matrices--For reasons that I now do not remember. However, many years ago, I wrote a short summary that is still available at this link
(In particular, they form a nice algebra, their characteristic polynomial has a closed form, etc. etc.)
A: Circulant Matrices? What is your motivation?
A: if $a > b$ then the matrix is a special case of a Robinsonian matrix or R-Matrix; I encountered that name  when searching for "matrix reordering" and remembered the question on MathOverflow
vis.pku.edu.cn/file/Summer%20School%202009/feket_matrix.pdf
A: I think the best description would be a Circulant Matrix, as the other "unknown (google)" suggests since what you are seeing is the vector (a b .... b) shifting right 1 on each row.
