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Somewhat unrelated to the spectral theorem, but how does one visualize eigenvectors?


I know mathematically why an eigenvector/eigenvalue is significant, but I can't really understand the applications of it. Spectral theorem I believe has something to do with diagonalizable matrices, but I fail to see how these matrices make a visual difference


Is that statement covered in blue not accurate enough? I can have I = I * I * I, which is a uniform scaling.


I think this means every symmetric matrix have their eigenvectors represented as the rotation component and eigenvalues as the scaling component? How does this change in the case of non-symmetric matrix, what kind of transformations do the eigenvalues and eigenvectors represent?


What is the situation when two eigen vectors are same?


I'm still unsure how the spectral theorem implies that every symmetric matrix performs a non-uniform scaling transformation. Specifically, how does the fact that that A has orthonormal eigenvectors imply that? Is it because the eigenvectors are orthonormal, which represent a change in basis, and the fact that the eigenvalues are real allow for scaling?


When you say a scaling in non-uniform, does this mean it doesn't preserve the object's shape?


What are some intuitive geometric interpretations to understanding the eigenvalues of a symmetric matrix? One can view them as the scaling factors in a different set of axes, but I'm wondering if there are other ways to look at this.