holds too. Fix two non-zero elements $x$ and $y$ of $\mathbb{F}_{p^{n}}$ and let $u=x+y$. $$\begin{split}
    \sum_{z\in\mathbb{F}_{p^{n}}} \left(\sum_{c\in \mathbb{F}_{p^{n}}} \overline{m_{x,c}}m_{z,c}\right)\overline{m_{z,y}} &=\sum_{z\in \mathbb{F}_{p^{n}}} \left(\sum_{c\in \mathbb{F}_{p^{n}}} \zeta_p^{-f(x+c)}\zeta_p^{f(z+c)}\right)\zeta_p^{-f(z+y)} \\
    &=\sum_{c,z\in \mathbb{F}_{p^{n}}} \zeta_p ^{-f(x+c)+f(z+c)-f(z+y)}\\
    &=p^{n+s}\zeta_p ^{-f(x+y)}
    \end{split}$$ Now let $z=a+x$ and $c=b+y$. Then $$p^{n+s}=\sum_{a,b\in \mathbb{F}_{p^{n}}} \zeta_p ^{-f(x+y+b)+f(a+b+x+y)-f(a+x+y)+f(x+y)}$$ holds. Thus, $$p^{n+s}=\sum_{a,b\in \mathbb{F}_{p^{n}}} \zeta_p ^{D_aD_b f(u)}.$$

Let $\Delta_f(a)=\sum_{x \in \mathbb{F}_{p^n}} \zeta_p^{D_af(x)}$. $f$ is an $s$-plateaued function from $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_{p}$ if and only if $\sum_{a\in \mathbb{F}_{p^{n}}} \overline{\Delta_f(a)}\Delta_f(a)=p^{2n+s}$.

We have $$MM^*MM^*=p^{n+s}MM^*.$$Let $$N=MM^*.$$ Then $$\begin{split}
    (N)_{0,a}&=\sum_{x\in \mathbb{F}_{p^{n}}} \zeta_p^{f(x)-f(a+x)} \\
    &=\overline{\Delta_f(a)}\\
    \end{split}$$ and $$\begin{split}
    (N)_{a,0}&=\sum_{x\in \mathbb{F}_{p^{n}}} \zeta_p^{f(a+x)-f(x)} \\
    &=\Delta_f(a)\\
    \end{split}$$ Therefore $$\begin{split}
    (N^2)_{0,0}&=\sum_{a\in \mathbb{F}_{p^{n}}} \overline{\Delta_f(a)}\Delta_f(a)\\
    &=p^{n+s}(N)_{0,0}\\
    &=p^{2n+s}\\
    \end{split}$$

Now we will use our characterization to provide a simple construction of $s$-plateaued functions. If $A$ is an $m \times n$ matrix and $B$ is an $s \times t$ matrix, then the Kronecker product $A \otimes B$ is the $ms \times nt$ block matrix:

$${\displaystyle \mathbf {A} \otimes \mathbf {B} ={\begin{bmatrix}a_{11}\mathbf {B} &\cdots &a_{1n}\mathbf {B} \\\vdots &\ddots &\vdots \\a_{m1}\mathbf {B} &\cdots &a_{mn}\mathbf {B} \end{bmatrix}},}$$

Let $f: \mathbb{F}_{p^{n}} \to \mathbb{F}_{p} $ and $g : \mathbb{F}_{p^{m}} \to \mathbb{F}_{p}$ be $s_1$-plateaued and $s_2$-plateaued functions respectively. Let $M$ and $N$ be the matrices whose entries determined by $m_{x,y}=\zeta_p^{f(x+y)}$ and $n_{a,b}=\zeta_p^{g(a+b)}$. Let $P=M \otimes N$ be the 