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I'm having trouble getting the geometric interpretation of complex eigenvectors and eigenvalues of a rotation matrix. I am given A= \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix} The given eigenvalues are: D=\begin{bmatrix} 2i & 0 \\ 0 & -2i \end{bmatrix} and the eigenvectors are:\begin{bmatrix} i & -i \\ 1 & 1 \end{bmatrix} I need to know the geometric interpretation of these complex eigenvectors and values and also those of $A^{\frac{1}{N}}$.
I know that A= $2\cdot\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = 2\cdot\begin{bmatrix}\cos{\pi /2} & -\sin{\pi /2} \\ \sin{\pi /2} & \cos{\pi /2}\end{bmatrix}$, so $A^{\frac{1}{N}}=2^{\frac{1}{N}}\cdot\begin{bmatrix}\cos{\pi /2N} & -\sin{\pi /2N} \\ \sin{\pi /2N} & \cos{\pi /2N}\end{bmatrix}$, which is a rotation of $\frac{\pi}{2N}$ counterclockwise and scaling by $2^{\frac{1}{N}}$.
I did read the earlier post about geometric interpretation, and the fantastic animated explanation, but I still don't get it. I apologise, it sounds like I'm asking for answers (I kind of am yeah), but I need a worked example to get anything done at all, and David C. Lay's textbook was incomprehensible to me.
Thank you very much for your help.
I'd be very grateful if someone could draw out the basis vectors, show me how I tell how much they rotate, and in which direction, and then how much it scales.

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  • $\begingroup$ So what is the question? $\endgroup$
    – user619894
    Mar 29, 2021 at 14:59
  • $\begingroup$ I'd like a geometric interpretation of $A^{\frac{1}{N}}$, its eigenvalues and eigenvectors, in image form, please. $\endgroup$
    – Memiya
    Mar 29, 2021 at 15:00
  • $\begingroup$ Do you want a figure with complex eigenvectors? Hard to draw, and if you don't like math.stackexchange.com/questions/241097/…, then I fear there is not much more that can be done. What exactly don't you follow there? $\endgroup$
    – user619894
    Mar 29, 2021 at 15:05
  • $\begingroup$ Yes please. I didn't get why the eigenvector just kept spinning. I tried to use it to think of how the basis vectors would look like but I can't. The eigenvectors given to me also have i on top and 1 on the bottom, so I can't translate those into basis vectors. $\endgroup$
    – Memiya
    Mar 29, 2021 at 15:06
  • $\begingroup$ We can't represent a complex number on the real axis, so we add an "imaginary" direction. So even a single complex number is represented by a 2d "vector" in the complex plane. Multiplying two complex numbers moves the position of this vector both by rotating and by stretching. Now if we have a complex vector, it is composed of of 2 of these "vectors" and each of them rotates in their own complex plane. $\endgroup$
    – user619894
    Mar 29, 2021 at 15:17

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