Using Hadamard Form of a Matrix in the Block Cipher Definition: A matrix A of size $2^n$ is a  Hadamard matrix, if has the following form 
$$
A=
\left(
\begin{array}{cc}
U &amp; V \\
V &amp; U
\end{array}
\right)_{2^n\times 2^n}\, ,
$$
where $U$ and $V$ are also Hadamard matrices. 

Example:  A Hadamard matrix A of size $2^2$ is as follows 
$$
A= \left( \begin {array}{cccc} a_{{0}}&amp;a_{{1}}&amp;a_{{2}}&amp;a_{{3}}
\\ a_{{1}}&amp;a_{{0}}&amp;a_{{3}}&amp;a_{{2}}
\\ a_{{2}}&amp;a_{{3}}&amp;a_{{0}}&amp;a_{{1}}
\\ a_{{3}}&amp;a_{{2}}&amp;a_{{1}}&amp;a_{{0}}\end {array}
 \right)\, ,
$$
Where $a_i$, $0\leq i \leq 3$ are from any arbitrary field. 

Consider B is a $2^n\times 2^n$ matrix such that the $(i,j)$the entry of B, denoted B$[i,j]$, is defined by B$[i,j]=a_{i \oplus j}$ where $i,j \in \{0,1,\cdots , 2^{n}-1\}$ and $i \oplus j$ means Xor between two numbers $i$ and $j$(for example $2 \oplus 3=1$ since $10 \oplus 11=01$).

Example: A matrix B of size $2^3$ is 
$$
B=
 \left( \begin {array}{cccccccc} a_{{0}}&amp;a_{{1}}&amp;a_{{2}}&amp;a_{{3}}&amp;a_{{4
}}&amp;a_{{5}}&amp;a_{{6}}&amp;a_{{7}}\\ a_{{1}}&amp;a_{{0}}&amp;a_{{3}}
&amp;a_{{2}}&amp;a_{{5}}&amp;a_{{4}}&amp;a_{{7}}&amp;a_{{6}}\\ a_{{2}}&amp;a
_{{3}}&amp;a_{{0}}&amp;a_{{1}}&amp;a_{{6}}&amp;a_{{7}}&amp;a_{{4}}&amp;a_{{5}}
\\ a_{{3}}&amp;a_{{2}}&amp;a_{{1}}&amp;a_{{0}}&amp;a_{{7}}&amp;a_{{6}}&amp;a
_{{5}}&amp;a_{{4}}\\ a_{{4}}&amp;a_{{5}}&amp;a_{{6}}&amp;a_{{7}}&amp;a_{
{0}}&amp;a_{{1}}&amp;a_{{2}}&amp;a_{{3}}\\ a_{{5}}&amp;a_{{4}}&amp;a_{{7
}}&amp;a_{{6}}&amp;a_{{1}}&amp;a_{{0}}&amp;a_{{3}}&amp;a_{{2}}\\a_{{6}}
&amp;a_{{7}}&amp;a_{{4}}&amp;a_{{5}}&amp;a_{{2}}&amp;a_{{3}}&amp;a_{{0}}&amp;a_{{1}}
\\ a_{{7}}&amp;a_{{6}}&amp;a_{{5}}&amp;a_{{4}}&amp;a_{{3}}&amp;a_{{2}}&amp;a
_{{1}}&amp;a_{{0}}\end {array} \right)=a_{i \oplus j}\qquad 
0\leq i,j \leq 7 \, .
$$

My question: Why the matrix B=$(a_{i \oplus j})$, $0\leq i,j \leq 2^{n}-1$, is an $2^n\times 2^n$ Hadamard matrix?

My Try: I can proof it by induction on $n$, but I would like to see its proof with other method. 

Background: When entries of a Hadamard matrix come from the finite field $GF(2^q)$ and be an MDS matrix, then the Hadamard matrix can be used in the diffusion layer of block cipher such as Khazad block cipher
This is evident. Let $B$ be a $2^n \times 2^n$ matrix. We have:


  $$a_{i\oplus (j+2^{n-1})}=a_{(2^{n-1}+i)\oplus j},a_{i\oplus j}=a_{(2^{n-1}+i)\oplus (2^{n-1}+j)}.$$


So, $B$ has the following form and therefore is a Hadamard matrix as follows
$$B=
\left(
\begin{array}{cc}
U &amp; V \\
V &amp; U
\end{array}
\right)_{2^n\times 2^n}\, $$