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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!