# Kernel matrix decomposition

I had a look at the sklearn.kernel_approxiamtion.Nystroem implementation, which is also described in this post:

Nystroem Method for Kernel Approximation

Here, a kernel-gram-matrix $$A$$ is decomposed using SVD to calculate $$A^{-1/2}$$.

I understand that A can be expressed as $$U\Sigma V^T$$ and $$A^{-1}= U\Sigma^{-1} V^T$$ . Also, I see that $$U=V$$ because $$A$$ is symmetric and thus, this is the same as eigendecomposition.

Now coming to my question: Here, $$A^{-1/2}$$ is calculated as $$A^{-1/2} = U\Sigma^{-1/2}V^T$$. I don't really understand this approach and would calculate $$A^{-1/2} = U\Sigma^{-1/2}$$, instead $$(A=(U\Sigma^{-1/2})(\Sigma^{-1/2}V^T)^T=(A)(A^{-1/2})^T)$$. I see that that $$VV^T=I$$ and so, in both cases $$A=A^{-1/2}(A^{-1/2})^T$$ is the same. So, why do I need $$V^T$$ in this equation, and how does it impact on my matrix $$A^{-1/2}$$?