Let $F=f\circ \bf{L}$, where $\bf{L}$ is a Linear transformation with matrix $(c^{i}_{j})$ of $dim=n\times n$ with $i$ for rows, and $j$ for columns. $F$ and $f$ are $C^2$ real-valued functions. We know that
The 2nd order partial derivatives of $F$ are $\displaystyle F_{ij}=\sum_{k,l=1}^{n}\frac{\partial^2 f}{\partial x_{k}\partial x_{l}}c^{k}_{i}c^{l}_{j}$ for $i,j=1,...,n$.
Now assuming that L is orthogonal, I'm asked to prove that $F_{11}+\dots+F_{nn}=f_{11}+\dots+f_{nn}$ (these are also 2nd order partial derivatives).
If L is orthogonal, then, with $l_i$ being the $i$-th column of $L$, $\displaystyle L^t L= I\Leftrightarrow l_i \bullet l_j= \begin{matrix}1, i=j \\ 0,i\neq j \end{matrix} \Leftrightarrow \sum_k c^{k}_{i}c^{k}_{j}=\begin{matrix}1, i=j \\ 0,i\neq j \end{matrix}$,
and $L L^t=I\Leftrightarrow \sum_k c^{i}_{k}c^{k}_{j}=\begin{matrix}1, i=j \\ 0,i\neq j \end{matrix}$ . (I haven't found useful this last condition, but I put it here, because one never knows when it might give someone an idea on how to solve my problem)
$\displaystyle \sum_i F_{ii}=\sum_{i,k,l}f_{l,k}c^{k}_{i}c^{l}_{i}$
Now assuming that $L$ is symmetric, $\displaystyle \sum_{i,k,l}f_{l,k}c^{k}_{i}c^{l}_{i}=\sum_{i,k,l}f_{l,k}c^{i}_{k}c^{i}_{l}$, which is equal to $\displaystyle f_{l,k}$ whenever $k=l$ and zero otherwise, by the first orthogonality condition deduced previously.
However, without assuming that $L$ is symmetric, I cannot prove this invariance of the laplacian. The exercise does not require/(allow to assume) symmetry. Where have I gone wrong?
Any help is appreciated. ;)
