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https://youtu.be/PFDu9oVAE-g?t=4m11s

a matrix can change the direction a vector is pointing.

However, some vectors don't get their directions changed, but instead are scaled.

It was described that those vectors are eigen vectors and because every other vector is moving, and the eigen vector isn't, the eigenvector can be seen as the axis of rotation for the matrix.

The visual representation of what the matrix is doing, "rotating around the eigen vector" is shown literally. However, I know for a fact matrices can have multiple eigenvectors, but how can something have multiple axis of rotations?

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  • $\begingroup$ Please don’t link to videos as context for your question. Include all of the relevant information in your question: What space are you working in? What matrix are you asking about? etc. $\endgroup$ – amd Sep 19 '18 at 20:31
  • $\begingroup$ rotations are tricky in the sense that in this context we are looking at a real matrix that has some non-real (complex) eigenvalues and eigenvectors. I'm not sure how to "visualize" such eigenvectors, but you might ask Grant on his youtube channel and who knows he might come up with another astonishing way of visualizing it. If it helps, think of a 2D rotation (2x2 matrix) of e.g. 90 degrees, that one won't have any real eigenvalues/eigenvectors, but still has two eigenvalues and eigenvectors, both with non-real values in them. $\endgroup$ – Keivan Nov 29 '18 at 19:56
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Leaving aside adjoined vector cases, things about 3×3 matrices over real space is like this.

Every matrix has a characteristic polynomial of 3rd order. This polynomial has 3 roots (eigenvalues). Due to the main theorem of algebra, either all of them are real, or one of them is real and other are 2 conjugated complex numbers.

In the first case, each of the 3 real eigenvalues $\lambda_i$ has corresponding eigenvector $v_i$ (direction). These directions are perpendicular, and transformation can be seen as scaling in each of the direction by the corresponding $\lambda_i$.

In case of only one real eigenvector $\lambda_1$, there only one corresponding eigenvector $v_1$ and for the pair of complex eigenvectors $\lambda_{2,3}$, there is a corresponding plane $p$ (perpendicular to the aforementioned eigenvector). The transformation is scaling by $\lambda_1$ in the direction of $v_1$, scaling by $|\lambda_{2,3}|$ in the plane $p$ and rotation around $v_1$ by angle $\arg \lambda_{2,3}$.

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