\begin{definition}[Definition:Combinatorial Matrix]
A '''combinatorial matrix''' is a square matrix specified as:
:$a_{i j} = y + \delta_{i j} x$
where $\delta_{i j}$ is the Kronecker delta.
In full, it can be written:
:$\begin {bmatrix}
x + y & y & \cdots & y \\
y & x + y & \cdots & y \\
\vdots & \vdots & \ddots & \vdots \\
y & y & \cdots & x + y
\end {bmatrix}$
It can also be expressed as:
:$x \mathbf I_n + y \mathbf J_n$
where:
:$\mathbf I_n$ is the unit matrix of order $n$
:$\mathbf J_n$ is the square ones matrix of order $n$.
\end{definition}