Proving that the orthogonal subspace is invariant

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SUMMARY

The discussion focuses on proving that the orthogonal subspace W^{\bot} is invariant under a unitary representation V when a subspace W is invariant under the action of a map D(g). The participants emphasize the importance of starting with a clear statement of the proof and leveraging the properties of unitary transformations. Specifically, they highlight that since W is invariant, the orthogonal complement W^{\bot} must also maintain this invariance due to the nature of orthogonal matrices, which are defined by their transpose being equal to their inverse.

PREREQUISITES
  • Understanding of unitary representations in linear algebra
  • Familiarity with orthogonal matrices and their properties
  • Knowledge of invariant subspaces
  • Basic concepts of linear transformations
NEXT STEPS
  • Study the properties of unitary transformations in detail
  • Learn about the proof techniques for invariant subspaces
  • Explore the relationship between orthogonal complements and linear transformations
  • Investigate applications of orthogonal matrices in quantum mechanics
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Mathematicians, physicists, and students studying linear algebra or quantum mechanics, particularly those interested in the properties of unitary representations and invariant subspaces.

Dixanadu
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Hi guys,

I couldn't 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 don't know how to apply that here. Can you guys help me out?

thanks!
 
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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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