[Math] non symmetric but diagonalizable matrix svd meaning

Question

I'm trying to figure out significance of SVD on different matrix classes (over the reals) and failing to understand the third case.

1. symmetric matrices : these matrices can be are orthogonally diagonalized, so the extra degree of freedom of rotation given in SVD is not needed.

2. non-symmetric non diagonizable matrices : The extra degree of freedom of rotation is exactly what is needed in order to show that the matrix essentially dilates an orthogonal set of vectors. This makes sense (to me), after all a linear operator is a simple operation so this single extra degree of freedom suffices.

3. non-symmetric that can however be diagonalized. In this case the diagonalization is over a non-orthogonal set of vectors. How come the extra degree of freedom of rotation in the SVD is enough in order to obtain the diagonlization over an orthogonal set? is there an intuitive geometric explanation for this?

Answer 1

I think you're looking for something that isn't there. In case 3, the diagonalization of your matrix has rather little to do with the SVD of the matrix. For example, the singular values are not functions of the eigenvalues.