Definition:Combinatorial Matrix

Definition
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 $n$.