# 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.

-
This kind of idea occurs in noncommutative algebra under the label "double centralizer theorem". For instance, you can find some applications in Lam's book on noncommutative rings. – KConrad Aug 27 '10 at 16:16
But... is this assertion always true? Suppose your field k is the complex number field $\mathbb{C}$. Doesn't exp(A) commute with all the matrices with which A commutes? – Qfwfq Aug 27 '10 at 17:07
That means, that $exp(A)$ can be written a polynomial in $A$. The point is that this polynomial may (and will) depend on $A$ itself. – Johannes Hahn Aug 27 '10 at 17:37
@unkown(google). Despite its appearance, exp(A) is a 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
Just to make it a little clearer, the reason that $exp(A)$ is a polynomial in $A$ is that $A$ satisfies its own minimal polynomial, so all terms of degree at least that of the minimal polynomial can be expressed as terms of lower degree, etc. – MTS Aug 27 '10 at 19:00

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.

-
Nice! I had no idea, thanks. That's the paper Tate published in 1966. I think that's kind of applications I'm looking for. Thank you again. – iravan Aug 27 '10 at 17:02
+1 for "Tate´s famous..." ;) – efq Aug 27 '10 at 17:22

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.

-
Unfortunately I don't know that much about functional analysis but I appreciate your comment very much. Thank you. – iravan Aug 27 '10 at 17:06
It is interesting to see an answer from the point of functional analysis after an answer involving abelian varieties over finite fields! – efq Aug 27 '10 at 17:25

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.

-