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I can understand the geometric meaning of $A^n$ (here $A \in R^{n \times n}$) when the eigenvalues of $A$ are all real. Basically, you scale up the each eigenvector $v_i$ along its direction by $\lambda_i$.
But what happens if the eigenvalues are complex? In that case, the eigenvectors will have complex elements too. I think I can sort to guess, $\lambda_i$ here does a bit of the rotation. But how do I define the direction of an eigenvector, with complex elements?
Perhaps a good example where to see what complex number has to do with real entries' matrices are of the kind $$\left(\begin{array}{cc}a&-b\\b&a\end{array}\right),$$ viewed as linear transformations $\mathbb R^2\to\mathbb R^2$.
Its characteristic polynomial is $\lambda^2-2a\lambda+a^2+b^2$ whose zeroes are $\lambda_{\pm}=\frac{2a\pm\sqrt{-4b^2}}{2}$ that is $$\lambda_+=a+ib,$$ $$\lambda_-=a-ib.$$ But the Eigen-Vectors would be $v_+=ie_1+e_2$ and $v_-=-ie_1+e_2$ which do not belong to $\mathbb R^2$.
However for the matrix above $$\left(\begin{array}{cc}a&-b\\b&a\end{array}\right)=r \left(\begin{array}{cc}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{array}\right),$$ with $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\frac{b}{a}$ and which allows the interpretation (for the transformation) of being an expansion by the $r$ factor and a rotation by an angle $\theta$ of the plane $\mathbb R^2$ which lets nothing invariant unless $b=0$.
