Looking for applications of a nice result in linear algebra - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T23:43:16Z http://mathoverflow.net/feeds/question/36893 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36893/looking-for-applications-of-a-nice-result-in-linear-algebra Looking for applications of a nice result in linear algebra iravan 2010-08-27T16:03:31Z 2010-08-27T18:00:01Z <p>Hello everybody</p> <p>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].\$</p> <p>I was wondering if anybody knows any (important) theorem which is proved using this result. Thank you.</p> http://mathoverflow.net/questions/36893/looking-for-applications-of-a-nice-result-in-linear-algebra/36896#36896 Answer by Franklin for Looking for applications of a nice result in linear algebra Franklin 2010-08-27T16:13:03Z 2010-08-27T16:13:03Z <p>That result sits inside a wider set of results. Search for spectral theorem, functional calculus of linear operators.</p> <p>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. </p> http://mathoverflow.net/questions/36893/looking-for-applications-of-a-nice-result-in-linear-algebra/36906#36906 Answer by Hunter Brooks for Looking for applications of a nice result in linear algebra Hunter Brooks 2010-08-27T16:46:36Z 2010-08-27T16:46:36Z <p>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.</p> http://mathoverflow.net/questions/36893/looking-for-applications-of-a-nice-result-in-linear-algebra/36907#36907 Answer by Andrew Stacey for Looking for applications of a nice result in linear algebra Andrew Stacey 2010-08-27T16:56:03Z 2010-08-27T16:56:03Z <p>This probably doesn't qualify as "important", but you put that in parentheses so I'll mention it anyway.</p> <p>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 <a href="http://www.math.ntnu.no/~stacey/Research/Preprints/coriemloop.html" rel="nofollow">the co-Riemannian structure of smooth loop spaces</a>.</p>