Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p^0 + p + \cdots + p^i)$, where $i \geq 0$.
Suppose that further the $(m,n)$-component $a_{m,n}$ of the matrix $A$ is defined as follows$\colon$ Now, we shall give the presentation of $\sigma$ in terms of the matrix in the following manner$\colon$ \begin{align}\label{MATRIX} & a_{1,1} = a_{2,1} = \cdots = a_{1 + p^0 + \cdots + p^k + 1,1} = 1 \phantom{A} {\mathrm{for}} \phantom{A} 0 \leq k < i \notag \\ & a_{2,1} = a_{4, 3} = \cdots = a_{m, m-1} = 1 \phantom{A} {\mathrm{for}} \phantom{A} m \not= 1 + p^0 + p + \cdots + p^k + 1, {\mathrm{where}}~0 \leq k < i \notag \\ & a_{1 + p^0 + \cdots + p^k + 1,1 + p^0 + p + \cdots + p^k + p^{k + 1}} = 1 \phantom{A} {\mathrm{for}} \phantom{A} 0 \leq k < i, \notag \\ & a_{m,n} = 0 \phantom{A} {\mathrm{otherwise.}} \end{align} where $a_{m,n}$ is the $(m,n)$-component of the matrix. For example in the case of $p = 2$ and $i = 2$, it is written as follows$\colon$
\begin{pmatrix}\label{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix}
