Let $A$ be a real symmetric positive definite $n \times n$ matrix. Then there exists some orthogonal matrix $O$ and diagonal matrix $D$ such that $O^T D O = A$, where $O$ has entries the eigenvectors of $A$ and $D$ has entries the real eigenvalues of $A$.
Let $x_0$ be some point in $\mathbb{R}^n$, and consider the vector $y = x_0 + O(x-x_0)$.
Then if I have any real twice-differentiable function $u: \mathbb{R}^n \to \mathbb{R}$, why can I write $u_{x_i} = \sum_{k=1}^n u_{y_k} o_{ki}$ and $u_{x_i x_j} = \sum_{k,l=1}^n u_{y_k y_l} o_{ki} o_{lj}$ where $o_{ki}$ is the entry in the $k$-th row and $i$-th column of the matrix $O$?
Also, what should be the geometric picture I have when I think about this vector $y$, and what part, if any, does the specific choice of $x_0$ play in these calculations?