Show that the intersection taken over the Gersgorin discs of all similar matrices of $A$ $=$ $\sigma (A)$

Question

Show that $\bigcap_S G(S^{-1}AS)$ $=$ $\sigma (A)$; the intersection is taken over all nonsingular $S$, and $\sigma (A)$ is the spectrum of $A$.

I'm lost as how to even begin to prove this fact. Any solutions, hints or suggestions would be appreciated.

Answer 1

Hint: we're allowed to use all non-singular $S$. One non-singular $S$ of particular interest is the $S$ that puts $A$ into its Jordan canonical form (or any upper-triangular form, if you prefer). For diagonalizable matrices, the Jordan form sufficient.

Otherwise, note that any Jordan block has the similarity $$ \pmatrix{ \lambda&1\\ &\lambda&1 \\ &&\ddots\\ &&&\lambda } \sim \pmatrix{ \lambda&1/n\\ &\lambda&1/n \\ &&\ddots\\ &&&\lambda } $$ (an analagous, explicitly constructed similarity can be applied to arbitrary upper-triangular matrices).