Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
Similar matrices have the same characteristic polynomial. –  Hans Lundmark Aug 26 '12 at 10:45
add comment

1 Answer

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.