Suppose that $f_n$ be the $n$th term of Fibonacci numbers. Consider the following special case of companion matrix $$ C_4:=\left( \begin {array}{cccc} 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 1&1&0&1 \end {array} \right) $$
With calculation I saw, when $n$ is an integer number ($n\in \mathbf{Z}$), then the $(2\,n)$th power of $C_4$ matrix, based on the Fibonacci number is in the following form
$$ C_4^{2n}= \left( \begin {array}{cccc} \left( f \left( n-1 \right) \right) ^{2 }&f \left( n-1 \right) f \left( n \right) &f \left( n-2 \right) f \left( n \right) &f \left( n-1 \right) f \left( n \right) \\ f \left( n-1 \right) f \left( n \right) &f \left( n-1 \right) f \left( n+1 \right) &f \left( n-1 \right) f \left( n \right) & \left( f \left( n \right) \right) ^{2} \\ \left( f \left( n \right) \right) ^{2}&f \left( n \right) f \left( n+1 \right) &f \left( n-1 \right) f \left( n+1 \right) &f \left( n \right) f \left( n+1 \right) \\ f \left( n \right) f \left( n+1 \right) &f \left( n \right) f \left( n+2 \right) &f \left( n \right) f \left( n+ 1 \right) & \left( f \left( n+1 \right) \right) ^{2}\end {array} \right) $$
In addition, the $(2\,n+1)$th power of $C_4$ matrix, by $C_4^{2n}$, is as shown
$$ C_4^{2n+1}= \left( \begin {array}{cccc} f \left( n-1 \right) f \left( n \right) & f \left( n-1 \right) f \left( n+1 \right) &f \left( n-1 \right) f \left( n \right) & \left( f \left( n \right) \right) ^{2} \\ \left( f \left( n \right) \right) ^{2}&f \left( n \right) f \left( n+1 \right) &f \left( n-1 \right) f \left( n+1 \right) &f \left( n \right) f \left( n+1 \right) \\ f \left( n \right) f \left( n+1 \right) &f \left( n \right) f \left( n+2 \right) &f \left( n \right) f \left( n+ 1 \right) & \left( f \left( n+1 \right) \right) ^{2} \\ \left( f \left( n+1 \right) \right) ^{2}&f \left( n+1 \right) f \left( n+2 \right) &f \left( n \right) f \left( n+2 \right) &f \left( n+1 \right) f \left( n+2 \right) \end {array} \right) $$
My question: How to find a closed-form expression for $C_4^n$ matrix when $n$ is an integer number.?
I would appreciate for any suggestions.
Edit(1): One of the interesting application of these closed-forms is obtaining new formula between Fibonacci terms. For instance, with $det(C_4^{2n})=1$ and $det(C_4^{2n+1})=-1$, we find two expression about Fibonacci numbers. For instance, assume that $f(n-2)=x$ and $f(n-1)=y$ then $$ \begin{array}{ccccc} f(n)=x+y & ,& f(n+1)=x+2\,y & , & f(n+2)=2\,x+3\, y \quad . \end{array} $$
Now, by replacing the values of $f(n-i)$, $-2\leq i \leq 2$, in the equation $det(C_4^{2\,n})=1$, we obtain the following equation based on the parameters $x$ and $y$ $$ {x}^{8}+4\,{x}^{7}y+2\,{x}^{6}{y}^{2}-8\,{x}^{5}{y}^{3}-5\,{x}^{4}{y}^ {4}+8\,{x}^{3}{y}^{5}+2\,{x}^{2}{y}^{6}-4\,x{y}^{7}+{y}^{8}=1 $$
which can be expressed by $$ \left( {x}^{2}+xy-{y}^{2} \right) ^{4}=1 $$
one solution of the above equation is $\left( {x}^{2}+xy-{y}^{2} \right) =1$, which is Cassini formula when $n$ is even , interesting.
Edit(2): Another application of $C_4$ matrix, is in the Fibonacci Coding Theory. Suppose that, we have a message matrix called $M$ of order $4$, as follows $$ M:=\left( \begin {array}{cccc} m_1&m_2&m_3&m_4\\ m_5&m_6&m_7&m_8\\ m_9&m_{10}&m_{11}&m_{12}\\ m_{13}&m_{14}&m_{15}&m_{16} \end {array} \right)\, . $$ where $m_i$,$1\leq i \leq 16$ are natural numbers. Now, you choose any natural numbers like $n$ and use $C_4^{2n}$ as an encoder matrix. The encoded matrix $E$ is obtained from the following equation
\begin{eqnarray} E&=&M \times C_4^{2n}\quad , \\ &=& \scriptsize{ \left( \begin {array}{cccc} m_1&m_2&m_3&m_4\\ m_5&m_6&m_7&m_8\\ m_9&m_{10}&m_{11}&m_{12}\\ m_{13}&m_{14}&m_{15}&m_{16} \end {array} \right) \, \left( \begin {array}{cccc} \left( f \left( n-1 \right) \right) ^{2 }&f \left( n-1 \right) f \left( n \right) &f \left( n-2 \right) f \left( n \right) &f \left( n-1 \right) f \left( n \right) \\ f \left( n-1 \right) f \left( n \right) &f \left( n-1 \right) f \left( n+1 \right) &f \left( n-1 \right) f \left( n \right) & \left( f \left( n \right) \right) ^{2} \\ \left( f \left( n \right) \right) ^{2}&f \left( n \right) f \left( n+1 \right) &f \left( n-1 \right) f \left( n+1 \right) &f \left( n \right) f \left( n+1 \right) \\ f \left( n \right) f \left( n+1 \right) &f \left( n \right) f \left( n+2 \right) &f \left( n \right) f \left( n+ 1 \right) & \left( f \left( n+1 \right) \right) ^{2}\end {array} \right)}\quad , \\ &=& \left( \begin {array}{cccc} e_1&e_2&e_3&e_4\\ e_5&e_6&e_7&e_8\\ e_9&e_{10}&e_{11}&e_{12}\\ e_{13}&e_{14}&e_{15}&e_{16} \end {array} \right)\, . \end{eqnarray}
By method of this article , it can be proved that there are the following relations between the entries of encoded matrix $E$:
\begin{equation} \left\{ \begin{array}{lcccc} \displaystyle{\frac{e_1}{e_2}=\frac{e_{5}}{e_{6}} =\frac{e_{9}}{e_{10}}=\frac{e_{13}}{e_{14}}}\approx \frac{1}{\varphi} &,&\\ \\ \displaystyle{\frac{e_2}{e_3}=\frac{e_{6}}{e_{7}} =\frac{e_{10}}{e_{11}}=\frac{e_{14}}{e_{15}}}\approx \varphi &,& &&(1)\\ \\ \displaystyle{\frac{e_3}{e_4}=\frac{e_{7}}{e_{8}}= \frac{e_{11}}{e_{12}}=\frac{e_{15}}{e_{16}}}\approx \frac{1}{\varphi} &.& \end{array}\right. \end{equation}
where $\varphi= \frac{1+\sqrt{5}}{2}$, is the Golden ratio. Notice that, the relation between entries of encoded matrix $E$ is independent of message matrix $M$. For example, consider the message matrix $M$ and encoder matrix $C_4^{40}$, be as follows $$ M=\left( \begin {array}{cccc} 93&32&68&55\\ 76&23&69&92\\ 54&74&99&31\\ 32&27&29&67 \end {array} \right) \quad , \quad C_4^{40}= \left( \begin {array}{cccc} 17480761&28284465&17480760&28284465\\ 28284465&45765226&28284465&45765225\\ 45765225&74049690&45765226&74049690\\ 74049690&119814915&74049690&119814916 \end {array} \right) $$ So, the encode matrix $E=M\times C_4^{40}$, is obtained in the following form $$ E= \left( \begin {array}{cccc} 9715581903&15720141722&9715581878&15720141745\\ 11949452536&19334620328&11949452529&19334620397\\ 9863309169&15959169509&9863309214&15959169466\\ 7611585662&12315804297&7611585659&12315804337 \end {array} \right)\tag{2} $$ we can check that the entries of the encoded matrix $(2)$, hold in the relation $(1)$. You can find the details of how to detect and correct errors in the received encoded matrix $E$, at the chapter four of this thesis.