Matrices send spheres to ellipsoids. Orthogonal matrices send spheres to spheres of the same radius.
Is the converse true? Is every matrix that sends spheres to spheres of the same radius an orthogonal matrix?
If so: Let $A$ be a matrix. Let $P$ be the set of vectors such that for all $p \in P, \|Ap\| = \|p\|$. According to the above, $A$ is orthogonal iff the complement of $P$ is empty. If $A$ is not orthogonal, can we somehow describe the "size" or nature of $P$ versus its compliment?