Linear algebraic dynamical system that has complex entries in the matrix Suppose that there is a dynamical system that has the form of $\mathbb{x}_{k+1} = A\mathbb{x}_k$.
Suppose that one eigenvalue of $A$ matrix is complex number, the form of $a-bi$.
We then convert $\mathbb{x}_k = P\mathbb{y}_k$ where matrix $P$ is the matrix of corresponding eigenvector of $a-bi$ eigenvalue. 
Then we would be able to write the dynamical system as the following: $\mathbb{y}_{k+1} = C\mathbb{y}_k$ where $C$ is \begin{bmatrix}a & -b \\b & a \end{bmatrix}
and then $A = PCP^{-1}$.
The question is, why does the eigenvalue of the matrix C equal to the eigenvalue of the matrix A?

Note that if $Av=\lambda v$ then $Cw=\lambda w$ where $w=P^{-1}v$. 
