# Can I calculate sum of squares (SSB and SSW) using the multivariate normal kernel?

I am new to the field of kernel analysis and I am trying to calculate sum of squares using multivariate normal kernel. So far I got

The multivariate normal kernel function:

$$f(x;H)=\frac{1}{n}\sum K_{H}(x-X_i)$$ where $$K(z)=ϕ(z)=(2π)^{-p/2} e^{-1/2 z'z}$$

Sum of Squares:

SSB=$$\sum N_{i}(ϕ(m_i)-ϕ(m))(ϕ(m_i)-ϕ(m))^T$$

SSW=$$\sum\sum (ϕ(x_{ij})-ϕ(m_{i}))ϕ(x_{ij})-ϕ(m_{i}))^T$$.

I tried to plug in $$f(x;H)$$ into the SSB and SSW, however, it gives me a complicated form that I am not able to simplify.

Is there a way to do this or is there any reference that can help me do it, because I can not find anything.