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Set of invertible matrices with real entries

  1. Dec 4, 2013 #1
    ##GL(n,\mathbb{R})## is set of invertible matrices with real entries. We know that
    [tex]SO(n,\mathbb{R}) \subset O(n,\mathbb{R}) \subset GL(n,\mathbb{R})[/tex]
    is there any specific subgroups of ##GL(n,\mathbb{R})## that is highly important.
     
    Last edited: Dec 4, 2013
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  3. Dec 4, 2013 #2

    jgens

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    You named most of the big ones. I would definitely add SL(n,R) to that list. Some of the subgroups consisting of diagonal matrices and say upper triangular matrices are important too.
     
  4. Dec 4, 2013 #3

    George Jones

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    Other important subgroups are the pseudo-orthogonal groups O(p,g) and SO(p,q) (e.g., the Lorentz group) and the symplectic groups.
     
  5. Dec 4, 2013 #4

    jgens

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    There is an embedding GL(n,C) < GL(2n,R) so all the important subgroups of GL(n,C) like the unitary groups can be regarded as subgroups of GL(2n,R) as well.
     
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