What is the term for this type of matrix?

Question

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.

Answer 1

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.