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.



What is the term for this type of matrix?
When $b = 0$, we have an $n \times n$ symmetric [arrowhead matrix](https://en.wikipedia.org/wiki/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.


