Looking for applications of a nice result in linear algebra Hello everybody
There is a nice classical result in linear algebra: if $A, B$ are two matrices in $M_n(k),$ where $k$ is a field, and $B$ commutes with every element of $M_n(k)$ which commutes with $A$, then $B = f(A)$ for some polynomial $f(x)$ in $k[x].$
I was wondering if anybody knows any (important) theorem which is proved using this result. Thank you.
 A: Tate's famous "Endomorphisms of Abelian Varieties over Finite Fields," which proves the Tate conjecture in the finite field case, uses the full force of the theorem of bicommutation in a reduction lemma. As KConrad mentions in the comments, the result you've cited is the special case of this theorem where one works with the subalgebra generated by one element.
A: That result sits inside a wider set of results. Search for spectral theorem, functional calculus of linear operators.
Books could be 
 Halmos, A Hilbert Space problem book
 if you also need to read more about linear operators in general I think in 
 Conway's Functional Analysis there is also stuff about these results, together with an introduction to functional analysis. 
A: This probably doesn't qualify as "important", but you put that in parentheses so I'll mention it anyway.
I used that result when figuring out some basic facts about polynomial loops in a compact, connected Lie group which I needed for my paper the co-Riemannian structure of smooth loop spaces.
