How to derive the three matrices of SVD from eigenvalue decomposition in Kernel PCA? Kernel PCA is usually done via eigenvalue decomposition of the Kernel Matrix $\mathbf{K}$ and standard PCA via SVD of the input $\mathbf{X}$.
In standard PCA as far as I know we can derive  $\mathbf{S}$ and $\mathbf{U}$ via two eigenvalue decompositions, of the Gram and Covariance/Correlation matrices:
$$
\begin{array}{c}
X=U\Sigma V^T\\
C=\dfrac{X^TX}{N-1}\\
G=\dfrac{XX^T}{N-1}\\
C=VE_CV^T\\
G=UE_GU^T\\
S=\sqrt{E_C(N-1)}\\
K=U_KE_KU^T\\
?=VE_?V^T
\end{array}
$$
But how does one get $\mathbf{V}$ in the case of a kernel? All posts I've ever read only discuss $\mathbf{U}$
Note:
I've read that $\mathbf{XV}=\mathbf{U\Sigma}$, however this relationship doesn't seem to hold for numpy.linalg.svd or scipy.linalg.svd

Based on @whuber's answer it's simply:
$$
\mathbf{V}=\mathbf{X^TU\Sigma^{-1}}
$$
