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Adjoint criterion for an invariant orthogonal complement

For a subspace

U

of a finite-dimensional inner-product space,

T (U) \subseteq U ⟺ T^{*} (U^{⊥}) \subseteq U^{⊥} .

This follows directly by moving

T

across the inner product and using

(U^{⊥})^{⊥} = U

.

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