Consider two vectors $\mathbf u,\mathbf v$. Let $\theta_0$ be the angle between them. Now multiply them by some matrix $M$. Let $\theta_1$ be the angle between $M\mathbf u$ and $M\mathbf v$. W.L.O.G assume $\theta_0>\theta_1$.
Now let $J$ be the matrix of eigenvalues and Jordan blocks for $M$. Let $\mathbf u',\mathbf v'$ be the vectors $\mathbf u,\mathbf v$ transformed into coordinates according to the basis of vectors in the Jordan normal form for $M$. Let $\theta_0'$ be the angle between these vectors (in the coordinate transformed space). Not consider the vectors $J\mathbf u'$ and $J\mathbf v'$. Let $\theta_1'$ be the angles between these vectors. Is it then the case that $\theta_0'>\theta_1'$ necessarily?
Question: Are we guaranteed that $\theta_0>\theta_1$ if and only if $\theta_0'>\theta_1'$? In other words: does a linear transformation preserve the ordering of angles between vectors?
My thinking is that a linear transformation can deform the space in various ways, but it shouldn't "bring vectors closer together" in the standard basis, while "moving them further apart" in the coordinate basis.
Can someone either prove this or provide a counterexample?
Clearly an arbitrary matrix need not preserve the angle between vectors. It could also, say, flip the ordering a vectors, etc. But if it brings two vectors closer together, does it also do so in relative to its Jordan normal form coordinate basis?
No, there is no reason for this to be true. Basically, an arbitrary linear change of coordinates does not respect angles at all, it only preserves linear independence.
For example, consider matrices with Jordan normal form $\begin{pmatrix}0&1&0\\0&0&0\\0&0&1\end{pmatrix}$
Let $v_1,v_2,v_3$ be the corresponding basis vectors, so that $F(v_2)=v_1$ and $F(v_3)=v_3$. Depending on $v_1,v_2,v_3$, you can get the angles between $v_2$ and $v_3$ and between $F(v_2)$ and $F(v_3)$ to be more or less anything you like (as long as it is possible for three linearly independent vectors in an Euclidean space), but the "angles" in the Jordan coordinate systems will always be $\frac\pi2$.