What is the term for this type of matrix? Is there an established term for the following type of square matrices?
$\begin{pmatrix}
c & c & c & c & \cdots & c & c \\
c & a & b & b & \cdots & b & b \\
c & b & a & b & \cdots & b & b \\
c & b & b & a & & b &  b \\
\vdots & \vdots & \vdots &   & \ddots & & \vdots \\
c & b & b & b & & a & b \\
c & b & b & b & \cdots & b & a \\
\end{pmatrix}$
The matrix contains just 3 different items $a, b, c$:


*

*The first row is $c$.

*The first column is $c$.

*The diagonal is $a$, except for the upper left corner.

*The remaining items are $b$.


Background: $a, b, c$ can be chosen such that the matrix is orthogonal, but has a constant first row. If the dimension is a square (i.e. the matrix is a $r^2 \times r^2$ matrix) then it is possible to choose all entries to be integers - up to a common (usually irrational) factor in front of the matrix for normalization.

When $b = 0$, we have an $n \times n$ symmetric arrowhead matrix. When $b \neq 0$, we have
$$\begin{bmatrix}
c & c & c & \cdots & c & c \\
c & a & b & \cdots & b & b \\
c & b & a & \cdots & b & b \\
c & b & b & \cdots & b & b \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
c & b & b & \cdots & a & b \\
c & b & b & \cdots & b & a \\
\end{bmatrix} = \begin{bmatrix}
c-b & c-b & c-b & \cdots & c-b & c-b \\
c-b & a-b & 0 & \cdots & 0 & 0 \\
c-b & 0 & a-b & \cdots & 0 & 0 \\
c-b & 0 & 0 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
c-b & 0 & 0 & \cdots & a-b & 0 \\
c-b & 0 & 0 & \cdots & 0 & a-b \\
\end{bmatrix} + b \, 1_n 1_n^{\top}$$
which is the sum of a symmetric arrowhead matrix and a (nonzero) multiple of the all-ones matrix.
