The orthogonal complement of an invariant subspace under a unitary operator is invariant

Let be an Inner product space with invariant subspace under unitary endomorphism . Then the Orthogonal complement is also invariant under . #m/thm/linalg

Proof

Let be an invariant subspace under . Then is also invariant under , and thus for any and

as required.

This extends to a Unitary representation of a finite group easily. Since Every finite complex representation of a compact group is equivalent to a unitary representation, this doesn't hold iff a representation is not unitary and non-finite.

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