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Proving that the orthogonal subspace is invariant

  1. Oct 10, 2013 #1
    Hi guys,

    I couldnt fit it all into the title, so here's what I'm trying to do. Basically, I have a unitary representation [itex]V[/itex]. There is a subspace of this, [itex]W[/itex], which is invariant if I act on it with any map [itex]D(g)[/itex]. How do I prove that the orthogonal subspace [itex]W^{\bot}[/itex] is also an invariant subspace of [itex]V[/itex]?

    I know that an orthogonal matrix is one where its transpose is its own inverse, but I dont know how to apply that here. Can you guys help me out?

    thanks!
     
  2. jcsd
  3. Oct 10, 2013 #2

    Office_Shredder

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    There's only one way that you can apply it really - you should start by writing out explicitly what it is you need to prove, and then use the fact that you have a unitary transformation.
     
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