Does there exist a linear transform from $\mathbb{R}^2\rightarrow\mathbb{R}^2$ that preserves the $\ell^3$ norm?
It is immediately obvious that combinations of permutations and change of sign of individual coordinates satisfy it.
I am wondering if there are more linear transforms that preserves the norm.
Thanks in advance.
Given a linear transformation $\,(x,y)\to (X,Y) := (a x+b y,c x+d y)\,$ then the condition for preserving $\,|x|^3+|y|^3\,$ splits into cases depending on the signs of $\,x\,$ and $\,y.\,$ and also $\,X\,$ and $\,Y.$ For the case $\,x>0,\,y>0,\,X>0,\,Y>0$ we must have $$ a^3+c^3=b^3+d^3=1, \quad a\,b^2+c\,d^2=a^2 b+c^2 d=0.$$ The two real solutions of this system of equations is $\,(X,Y) = (x,y)\,$ and $\,(X,Y) = (y,x).$
The other cases are similar and allows to negate $\,X\,$ or $\,Y.$