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.

isa polynomial on A. That is, for every A, you can find a polynomial r(t) such that exp(A)= r(A). It is the Lagrange-Sylvester interpolation polynomial: see the book of Gantmacher, The theory of matrices, books.google.es/… . – a.r. Aug 27 '10 at 17:49