Is there a name for a square matrix with constant diagonal and off-diagonal elements?

Question

I am interested in real symmetric matrices of the form:

$$\mathbf{M} = \begin{bmatrix} a & t & t & \cdots & t \\ t & a & t & \cdots & t \\ t & t & a & \cdots & t \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ t & t & t & \cdots & a \text{ } \text{ } \\ \end{bmatrix}.$$

This is a simple matrix form where the diagonal elements have a constant value $a \in \mathbb{R}$ and the off-diagonal elements have a constant value $t \in \mathbb{R}$. Some useful special cases of this matrix form are the centering matrix and the equicorrelation matrix. (This matrix form is also a particular case of the Toeplitz matrix, but it is much simpler than that general form.)

Question: Does this matrix form have a name?

Answer 1

In old statistics literature, these are sometimes called completely symmetric matrices. I have made a light effort to revive the term in my writing or teaching (which is usually a combinatorial context).

Answer 2

This class of matrices is now examined in detail in O'Neill (2020) where they are called "double-constant" matrices. Although these matrices have been mentioned in some books on linear algebra in statistics, they do not appear to have had any existing name in the literature prior to this paper. As noted in the paper, this class of matrices includes the centering matrix and the equicorrelation matrix, both used widely in statistical applications.