# 6 orthonormal subspace question

1. Oct 28, 2011

### nhrock3

6)there is normal T in unitarian final space.
$$v\neq0,v\in V$$ prove that if $$\{sp(v)\}^{\perp}$$ is T variant then
v is eigenvector of T
?
hint:prove that $$T*(v)$$ is orthogonal to $$\{sp(v)\}^{\perp}$$
what i have done:
suppose $$u\in\{sp(v)\}^{\perp}$$
we take the definition of $$T*$$
(Tu,v)=(u,T*v)
(Tu,v)=0 because $$\{sp(v)\}^{\perp}$$ is T variant so T*v is orthogonal
to u.
i know also
V=$$\{sp(v)\}^{\perp}\oplus\{sp(v)\}$$
what to do next ,how to prove the actual question?