Kernel PCA is usually done via eigenvalue decomposition of the Kernel Matrix **K** and standard PCA via SVD of the input **X**. We can derive **S** and **U** from eigenvalue decomposition 

![enter image description here][1] 

![enter image description here][2]

How can I get the matrix **V**? I need it for certain calculations like biplots.

In standard PCA as far as I know we can derive **U** and **V** via two eigenvalue decompositions, of the Gram and Covariance/Correlation matrices:

![enter image description here][3]

![enter image description here][4]

![enter image description here][5]

![enter image description here][6]

[![enter image description here][7]][7]


  [1]: https://i.sstatic.net/IA9SZ.png
  [2]: https://i.sstatic.net/xYXNG.png
  [3]: https://i.sstatic.net/55puB.png
  [4]: https://i.sstatic.net/SzBJk.png
  [5]: https://i.sstatic.net/09QvC.png
  [6]: https://i.sstatic.net/fDNYe.png
  [7]: https://i.sstatic.net/uS802.png