1. Limited time only! Sign up for a free 30min personal 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!

Matrix groups, GL_2(C)

  1. Sep 24, 2011 #1
    1. The problem statement, all variables and given/known data

    [itex]A= \left( \begin{matrix}
    i & 0 \\
    0 &-i
    \end{matrix} \right) [/itex]
    , [itex]B= \left( \begin{matrix}
    0 & 1 \\
    1 & 0
    \end{matrix} \right) [/itex]
    \\
    Show that [itex]\langle A, B \rangle[/itex] is subgroup of [itex]GL_2(\mathbb{C})[/itex]. And Show that [itex]\langle A, B \rangle[/itex] generated by [itex]A[/itex] and [itex]B[/itex], and order of [itex]\langle A, B \rangle[/itex] is 8 ?

    2. Relevant equations

    [itex] GL_2(\mathbb{C}) = \big\lbrace X \in M_2(\mathbb{C}) ~~\vert ~~ \exists Y\in M_2(\mathbb{C}) ~ with~ XY=YX=I \big\rbrace[/itex] \\
    which [itex] Y[/itex] is inverse of [itex] X[/itex]

    3. The attempt at a solution
     
  2. jcsd
  3. Sep 24, 2011 #2
    And what have you done?? What do you have to do to show something is a subgroup?
     
  4. Sep 24, 2011 #3
    [itex]\langle A, B \rangle = \left( \begin{matrix} 0 & i \\ -i & 0 \end{matrix} \right)[/itex] and det(<A,B>)=-1, hence det(<A,B>) in [itex]GL_2(\mathbb{C})[/itex]. right?

    on the other hand, if we want to show [itex]\langle A, B \rangle [/itex] generated by A and B,
    we need to show that A and B are linear independent ???
     
    Last edited: Sep 24, 2011
  5. Sep 24, 2011 #4
    I am confused, because
    [itex]\langle A, B \rangle[/itex][itex] = \Big\langle \left( \begin{matrix} i & 0 \\ 0 & -i \end{matrix} \right) \left( \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right)\Big\rangle [/itex][itex]= \left( \begin{matrix} 0 & i \\ -i & 0 \end{matrix} \right) [/itex] right?

    and this is just an element of [itex]GL_2{\mathbb{C}}[/itex], not a group of [itex]GL_2{\mathbb{C}}[/itex], right?????
     
    Last edited: Sep 24, 2011
  6. Sep 24, 2011 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Right. I'm not sure what <A,B> is supposed to mean, but I think you just supposed to check that the group generated by all possible products of A and B is a subgroup of order 8.
     
  7. Sep 24, 2011 #6
    So , should I try to find all possible products of A and B , or is there some trick to find it?
     
  8. Sep 24, 2011 #7

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Well, A^4=I and B^2=I, right? Showing AB=(-BA) would also help a lot.
     
  9. Sep 25, 2011 #8
    I try to calculate possibilities,

    I, A, B, AB, BA, AAB, AAA, BAA

    and there are 8 elements , is it the answer?
     
  10. Sep 25, 2011 #9

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    The answer would be a PROOF that those 8 elements form a group. Just listing them isn't enough. Besides, I don't think all of those are different. Isn't AAB=BAA?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Matrix groups, GL_2(C)
  1. Matrix groups (Replies: 8)

  2. Matrix groups (Replies: 2)

Loading...