I'm using a lot of matrices that look like this: $$A_3 = \begin{bmatrix} a & b & b\\ b & a & b\\ b & b & a \end{bmatrix} $$

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!