# Proving that the orthogonal subspace is invariant

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 $V$. There is a subspace of this, $W$, which is invariant if I act on it with any map $D(g)$. How do I prove that the orthogonal subspace $W^{\bot}$ is also an invariant subspace of $V$?

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!

## Answers and Replies

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Office_Shredder
Staff Emeritus
Science Advisor
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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.