Is there a procedure to locally linearize nonlinear operators from $\Bbb R^n$ to $\Bbb R^n$?

Question

If I have a nonlinear operator $T: \mathbb R^n \to \mathbb R^n$ such that $\| T(x) \| = \| x \|$ for each $x$ in $\mathbb R^n$, is there a procedure to locally approximate it with a linear orthogonal operator $A : \mathbb R^n \to \mathbb R^n$ inside the neighborhood of a certain vector $x$, for each vector $x$ in $\mathbb R^n$?

Answer 1

The object that does this kind of approximation is the gradient of $T$ (the precise name is probably “Jacobian” since we are working with vector-valued functions). More precisely, if you want to approximate $T$ around some vector $x_0$, then $$ T(x)\approx T(x_0)+\nabla T(x_0)\cdot (x-x_0). $$

In your hypothesis, though, the gradient might not be orthogonal. Think about it in this way: if you only impose $\|T(x)\|=\|x\|$, this means that your operator $T$ acts on the spherical surfaces in $\mathbb R^n$ centered at $0$ with fixed radius, and preserves all these surfaces. But then, the way $T$ acts on such spheres is essentially arbitrary. But locally, these spherical surfaces look like $\mathbb R^{n-1}$… This means that the linear approximation of $T$ restricted to such spheres is essentially an arbitrary linear operator, thus not orthogonal on $\mathbb R^{n-1}$. This suggests that the full approximation of $T$ is not in general orthogonal (if an operator is orthogonal, it must also be an orthogonal operator on all the linear subspaces of the space it acts on).

Edit.

If you want another bit of intuition: orthogonal linear maps preserve angles between lines, thus a map that is approximated by a linear orthogonal map at a point $x$ should somehow preserve the angles between curves that pass through $x$ (this is true an can in fact be made rigorous). If you think for some time, maybe you can think of a counterexample of this in the hypotheses you assumed (say, in $\mathbb R^2$).

Small remark: of course, I am assuming the norm you are using is the Euclidean norm, so that the level sets of the norm are spherical surfaces.