I have often seen it stated that a symmetric real-valued matrix $A$ acts on the unit circle $x^T x = 1$ by mapping it to $x^TAx = 1$, and that the resulting ellipsoid (which is defined by such quadratic forms) has as its principal axes the eigenvectors of $A$, with lengths related to the corresponding eigenvalues.
For a two-dimensional ellipse where $A$ involves no rotation, such a statement can be proved because the axes of the ellipse are defined/proved to be what they are in terms of classical Euclidean plane geometry. However, I have never seen a proof of the general $n$-dimensional case, rather it is defined that the principal axes of the ellipse are defined by the eigenvectors.
So if one were to take the statement that "the principal axes of an ellipsoid are the eigenvectors of $A$" as a definition, in what way can one corroborate this geometrically?