Linear Algebra Eigenbasis Proof Help

Question

Let T be the orthogonal projection on a subspace V of Rn. Proven T has an orthonormal eigenbasis. (I already proved this part with the eigenvalues of T being 0 and 1)

Let S be the reflection about a subspace V of Rn. Prove that S also has an orthonormal eigenbasis.

For this I have found S has the eigenvalues 1 and -1 because of a theorem which states "the possible real eigenvalues of an orthogonal matrix are 1 and -1."

I'm not sure where to go from here, please help solve the bolded part. :)

Answer 1

We can find an orthonormal eigenbasis explicitly in each case in the following way:

Let $v_1,\dots,v_d$ be an orthonormal eigenbasis of $V$. Extend this to an orthonormal eigenbasis $v_1,\dots,v_n$ of $\Bbb R^n$, noting that $v_{d+1},\dots,v_n$ are orthogonal to $V$.

Verify that these are eigenvectors of $S$ and $T$. Note the associated eigenvalues.