>>11176122Let C be the field of complex numbers, n>=1 an integer and if x,y are vectors in C^n, <x,y>:= sum_k=1^n y_k(x_k)* (<_,_> is the standard hermitian product). We define ||x|| := sqrt (<x,x>) for every x in C^n.
Let f:C^n -> C^n a linear map.Let W(f) be the set of complex numbers of the form <x,f(x)> where x ranges over the set of y such that ||y|| = 1.
Show that W(f) is convex.
NB: this is fairly obvious when f is an hermitian operator, but batshit crazy insane in the general case.
This infamous exercise was given in an oral entrance exam to a french top school (ENS ULM) without any indication.
for related concepts and a possible solution, look for "Hausdorffian of a matrix", or things like that.
