Define the operator $$ Pu=\sum_{ij} a_{ij}(x) \partial^2_{ij} u+\sum_k b_k(x) \partial_k u+c(x) u. $$ My question is, in what situation can we somehow "diagonalise" the principle part of $P$, namely $\sum_{ij} a_{ij}(x) \partial^2_{ij} u$?
That is, to find a system of coordinates $y_i$ such that $$ \sum_{ij} a_{ij}(x) \partial^2_{ij} u= \sum_{k} \lambda_k(x) \partial^2_{y_k\space y_k} u+\text{lower order derivatives of }u $$
I find that, although this is quite easy for constant $a_{ij}$, it is very difficult in the general case, since differentiaing leads to extra terms according to chain rule, when changing the coordinates.
I am just wondering if there exists an "appropriate" way of "diagonalising" the principle part, or to find a "canonical form".
According to this Wikipedia article it is possible for two dimensional elliptic pdes but in higher dimensions it is in general not possible.