I am trying to understand Singular Value Decomposition. I have not read this explicitly, but as far as I understand it we have:
$\mathbf{V}^\top$ changes the basis from the canonical/standard basis in $\mathbb{R}^n$ to the basis, for which the linear transformation $\mathbf{A}$ is merely a scaling and/or change of dimension.
I.e.,
The first step of SVD changes the basis of the domain to a basis such that in the resulting basis the linear transformation $\mathbf{A}$ is merely a scaling and/or change of dimension.
Is that true?
For reference, this is the definition I am working with (MML Deisenroth):
