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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?

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Similar matrices have the same characteristic polynomial. – Hans Lundmark Aug 26 '12 at 10:45

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

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