91Ó°ÊÓ

A table of numbers and symbols arranged in the form of rows and columns is known as matrix.

Consider the given matrix,

$C_4=\left[\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right]$

Hence, we compute,

$$\begin{gathered} C_4^2=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right] \\ {C}_4^3=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\end{array}\right] \\ {C}_4^4=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]=I_4 \end{gathered}$$

Therefore, the powers of$C_4^2,C_4^3,C_4^4,…$are,

$C_4^2=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right],C_4^3=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\end{array}\right],C_4^4=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]=I_4$

Solve using the determinant,

$$\begin{gathered} \det \left(C_4-\mathrm{λ}I\right)=0 \\ {\mathrm{λ}}^4-1=0 \\ {\mathrm{λ}}^2=Â\pm 1 \\ {\mathrm{λ}}_{1,2}=Â\pm 1,{\mathrm{λ}}_{3,4}=Â\pm i \end{gathered}$$

Thus, the eigenvectors are,$v_1=\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right],v_2=\left[\begin{array}{c}-1\\ 1\\ -1\\ 1\end{array}\right],v_3=\left[\begin{array}{c}1\\ -i\\ -1\\ i\end{array}\right],v_4=\left[\begin{array}{c}1\\ i\\ -1\\ -i\end{array}\right]$

Hence, the eigenbasis are, $\left\{v_{1,}{v}_2,v_3,v_4\right\}$

Therefore, the complex eigenvalues of C₄ are ${\mathrm{λ}}_{1,2}=Â\pm 1,{\mathrm{λ}}_{3,4}=Â\pm i$and the complex eigenbasis are $\left\{v_{1,}{v}_2,v_3,v_4\right\}$.

Consider the matrix,

$M=\left[\begin{array}{cccc}\mathrm{a}& \mathrm{d}& \mathrm{c}& \mathrm{b}\\ \mathrm{b}& \mathrm{a}& \mathrm{d}& \mathrm{c}\\ \mathrm{c}& \mathrm{b}& \mathrm{a}& \mathrm{d}\\ \mathrm{d}& \mathrm{c}& \mathrm{d}& \mathrm{a}\end{array}\right]=a{I}_4+b{C}_4+c{C}_4^2+d{C}_4^3$

Therefore, the matrix M is called circulant as it is in the form $M=\left[\begin{array}{cccc}\mathrm{a}& \mathrm{d}& \mathrm{c}& \mathrm{b}\\ \mathrm{b}& \mathrm{a}& \mathrm{d}& \mathrm{c}\\ \mathrm{c}& \mathrm{b}& \mathrm{a}& \mathrm{d}\\ \mathrm{d}& \mathrm{c}& \mathrm{d}& \mathrm{a}\end{array}\right]$.