Why are the axis of an ellipsoid eigenvectors?

The axes of this ellipsoid are (the multiples of) $e_1=(1,0,0,\ldots,0)$, $e_2=(0,1,0,\ldots,0)$, …, $e_n=(0,0,0,\ldots,1)$ (and the corresponding eigenvalues are $\lambda_1,\lambda_2,\ldots,\lambda_n$). But $\Lambda=P^{-1}AP$ and $P$ is a change-of-bases matrix. So, the columns of $P$ are eigenvectors of $A$ (and, again, the corresponding eigenvalues are $\lambda_1,\lambda_2,\ldots,\lambda_n$). This was just a change of basis, so, geometrically, the columns of $P$ are the still the vectors $e_1,\ldots,e_n$.