What are square matrices of this form called?

Is there a name for square matrices whose rows and columns (and only these; indeed, the main diagonal entries always sum to $$n \alpha,$$ where $$n$$ is the size of the matrix, and $$\alpha$$ each entry of the form $$a_{ii}$$) sum to the same value?

For example, something like $$\begin{bmatrix} 2&1&0&1\\ 1&2&1&0\\ 0&1&2&1\\ 1&0&1&2 \end{bmatrix}.$$

PS. As it might help, an additional property that these matrices have is that they are at least persymmetric (that is in addition to the constancy of the row/column-sum above), although the antidiagonal need not be constant.

• Not exactly what you asked, but a matrix with constant diagonals like your example is called a Toeplitz matrix. – Klaus Feb 15 at 17:55
• @Klaus I didn't know that. That would mean the matrices I am interested in are a subclass of Toeplitz matrices. It's some progress, if you ask me! – Allawonder Feb 15 at 18:08
• Maybe a "multiple of a doubly stochastic matrix"? – Teddan the Terran Feb 15 at 18:22
• @TeddantheTerran What's a doubly stochastic matrix? Could you give an example too? – Allawonder Feb 15 at 18:46
• @Allawonder No, but I meant that it is a multiple of a stochastic matrix: Divide all entries by 4, and call this matrix M. Then M is doubly stochastic, and 4M is your matrix. – Teddan the Terran Feb 15 at 21:59