Hadamard Form of a Circulant Matrix

Definition: Let a field $\mathbb{F}$. Consider an $2^n \times 2^n$ matrix $\bf H$ over $\mathbb{F}$. $\bf H$ is called Hadamard over $\mathbb{F}$ if and only if $${\bf H}=\left( \begin{array}{cc} {\bf U} & {\bf V} \\ {\bf V} & {\bf U} \end{array} \right)$$
where $\bf U$ and $\bf V$ are $2^{n-1} \times 2^{n-1}$ Hadamard matrices over $\mathbb{F}$.

Consider a circulant matrix ${\bf C}_4$ over $\mathbb{R}$ which is defined by $${\bf C}_4= \left( \begin {array}{cccc} 0&1&1&0\\ 0&0&1&1\\ 1&0&0&1\\ 1&1&0&0 \end {array} \right).$$ Let the $m$th power of ${\bf C}_4$ is denoted by ${\bf C}_4^m$ for some $m$. For instance, ${\bf C}_4^4$ and ${\bf C}_4^8$ are as follows

$$\begin {array}{cc} {\bf C}_4^4=\left( \begin {array}{cccc} 2&4&6&4\\ 4&2&4&6 \\6&4&2&4\\ 4&6&4&2\end {array} \right),& {\bf C}_4^8=\left( \begin {array}{cccc} 72&64&56&64\\ 64&72&64& 56\\ 56&64&72&64\\ 64&56&64&72 \end {array} \right). \end {array}$$

My Question: Let $n$ is a positive integer number. How to prove ${\bf C}_4^{2^n}$,$n>1$, is Hadamard over $\mathbb{R}$.

My try: I numerically checked ${\bf C}_4^m$ in general is a Toeplitz matrix. For instance, ${\bf C}_4^5$ and ${\bf C}_4^{10}$ are:

$$\begin {array}{cc} {\bf C}_4^5= \left( \begin {array}{cccc} 10&6&6&10\\ 10&10&6&6 \\ 6&10&10&6\\ 6&6&10&10 \end {array} \right), & {\bf C}_4^{10}= \left( \begin {array}{cccc} 256&240&256&272\\ 272&256&240&256\\ 256&272&256&240\\ 240&256&272&256\end {array} \right). \end {array}$$

The Jordan canonical form of ${\bf C}_4$ is in the following form:

$$\begin {array}{cc} {\bf Q}= \left( \begin {array}{cccc} 1/4&1/4&1/4&1/4\\ -1/4&1/4&-1/4\,i&1/4\,i\\ 1/4&1/4&-1/4&-1/4\\ -1/4&1/4&1/4\,i&-1/4\,i \end {array} \right), & {\bf J}= \left( \begin {array}{cccc} 0&0&0&0\\ 0&2&0&0\\ 0&0&-1-i&0\\ 0&0&0&-1+i \end {array} \right), \end {array}$$ where ${\bf C}_4={\bf Q}\cdot {\bf J} \cdot {\bf Q}^{-1}$ and $i=\sqrt{-1}$. It can be verified that ${\bf J}^{2^n}$ for $n> 2$ is: $${\bf J}^{2^n}= \left( \begin {array}{cccc} 0&0&0&0\\ 0&{2}^{2^n}&0&0\\ 0&0&{2}^{2^{n-1}}&0\\ 0&0&0&{2}^{2^{n-1}}\end {array} \right).$$ Now assume that $a=2^{\displaystyle{2^n-2}}$. Then we can checked that ${\bf C}_4^{2^n}$ is: $${\bf C}_4^{2^n}= \left( \begin {array}{cccc} a+\sqrt {a}&a&a-\sqrt {a}&a\\ a&a+\sqrt {a}&a&a-\sqrt {a}\\ a-\sqrt {a}&a&a+\sqrt {a}&a\\ a&a-\sqrt {a}&a&a+\sqrt {a} \end {array} \right).$$ Is there another simple proof (especially based on the circulant matrices ) for this question?

Thanks for any suggestions.

• Notice: $C_4^2$ is not hadamard – Exodd Jun 20 '18 at 12:36
• @Exodd The question is edited by your notice. Thanks – Amin235 Jun 20 '18 at 13:37

The circulant matrices form an algebra, meaning that given any polynomial $p\in \mathbb C[x]$, then $p(C)$ is still circulant. Notice moreover that a circulant matric is a particular case of Toeplitz matrix.
Let us prove that $C_4^{2^n}$ is symmetric if $n>1$. You already noticed that $S = C_4^4$ is symmetric, but then $S^k$ is symmetric for every $k$, so $C_4^{2^n} = S^{2^{n-2}}$ is symmetric if $n>1$.
Notice : With the same proof you prove that $C_4^{n}$ is Hadamard if and only if $n$ is a multiple of 4.
• I know what you mean. In fact, if $a(x)$ be the polynomial that is associated with the first row of an $n \times n$ circulant matrix $A$ then the $i$th row of $A$ is obtained by $x^i\, a(x)\mod{x^n-1}$ for $0 \leq i \leq n-1$. Thanks. – Amin235 Jun 20 '18 at 14:14