1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Connected components of upper triangular matrices

  1. Apr 18, 2006 #1
    Hello, i'm working on a problem in topology. I'm supposed to find the number of connected components of the group of 2x2 invertible upper triangular matrices over R which i shall call [itex]B_2[/itex].

    I've tried it a bit, but I don't know for sure if my approach (and answer) is correct.

    Since any homeomorphism preserves connectivity, I consider the trivial homeomorphism of [itex]B_2[/itex] to [itex]\mathbf{R}^3[/itex].

    Since the matrices have to be invertible the determinant is non-zero which means that for matrices (a b | 0 c), ac > 0 or ac < 0. But he piece with positive determinant splits in 2 non connected pieces, {(a,b,c) | a > 0 and c > 0} and {(a,b,c) | a < 0 and c < 0}. The same sort of thing holds for the piece with negative determinant.

    Is it correct to assert from this that [itex]B_2[/itex] has 4 connected components?
  2. jcsd
  3. Apr 18, 2006 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    I think that shows it splits into at least 4 connected components. You've not show that each of those individual pieces is connected.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Connected components of upper triangular matrices
  1. Triangular Matrices (Replies: 4)