Triangulability theorem: Difference between revisions

Revision as of 21:04, 2 September 2008

This article gives the statement, and proof, of a particular subgroup in a group being conjugate-dense: in other words, every element of the group is conjugate to some element of the subgroup

Statement

Let $k$ be an algebraically closed field. Denote by $GL_{n}(k)$ the general linear group of invertible $n\times n$ matrices over $k$ and by $B_{n}(k)$ the Borel subgroup: the subgroup of invertible upper triangular matrices. Then, $B(n,k)$ is conjugate-dense in $GL(n,k)$ . In other words, given any matrix in $GL(n,k)$ , we can conjugate it (or change basis) to make it upper triangular.

This is an immediate corollary of the Jordan canonical form theorem, though it can be proved independently as well.

Note that the result fails over non-algebraically closed fields; in particular, it fails when $k$ is a finite field.

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 ==Statement==
-Let <math>k</math> be an [[algebraically closed field]]. Denote by <math>GL(n,k)</math> the [[general linear group]] of invertible <math>n \times n</math> matrices over <math>k</math> and by <math>B(n,k)</math> the subgroup of invertible upper triangular matrices. Then, <math>B(n,k)</math> is conjugate-dense in <math>GL(n,k)</math>. In other words, given any matrix in <math>GL(n,k)</math>, we can conjugate it (or change basis) to make it upper triangular.
+Let <math>k</math> be an [[algebraically closed field]]. Denote by <math>GL_n(k)</math> the [[general linear group]] of invertible <math>n \times n</math> matrices over <math>k</math> and by <math>B_n(k)</math> the [[fact about::Borel subgroup in general linear group|Borel subgroup]]: the subgroup of invertible upper triangular matrices. Then, <math>B(n,k)</math> is conjugate-dense in <math>GL(n,k)</math>. In other words, given any matrix in <math>GL(n,k)</math>, we can conjugate it (or change basis) to make it upper triangular.
 This is an immediate corollary of the [[Jordan canonical form theorem]], though it can be proved independently as well.
 Note that the result fails over non-algebraically closed fields; in particular, it fails when <math>k</math> is a [[finite field]].