In a machine learning project I am working on, I have came across the following problem. I have a symmetric, $d\times d$ matrix $K_{oij}=k(x_i,x_j)$ where $k$ is a Gaussian kernel function. $x_1,\dots ,x_d$ are some data points. Then I have the following relation $K^{-1}_{o} = G^{T}G$. The inverse of $K_o$ is a symmetric matrix and we know that $G$ is a $d \times d$ matrix as well, since it is said in the paper I am following that the vector $\phi = Gk_o$ is a $d$ dimensional vector, where $k_o$ is a $d \times 1$ column vector, which implies that $G$ is $d \times d$.
Now, I need to calculate that $\phi$ vector and therefore I need the matrix $G$. But I only have $K_o$ and its inverse at hand, which is related to $G$ with $K^{-1}_{o} = G^{T}G$. I don't see an easy way to calculate $G$ and became blocked. Is there a way to solve that kind of equation or am I missing something here?