What are the eigenvalues of matrix $I-uu^T$ where where $u$ is a unit vector and $I$ is the identity matrix in $\mathbb R^n$

Question

I have tried to find an eigenvalue $0$, by $(I-uu^T)u=u-1u=0u$. I have felt that others are $1$,but that's no really reliable.

Answer 1

Put $\;U:=Span\{u\}\;$ and denote by $\;\langle .\rangle\;$ the usual inner product, then for any $\;w\in U^\perp\;$ we get:

$$(I-uu^t)w=w-u(u^tw)=w-\langle w,u\rangle u=w\implies w$$

is an eigenvector with eigenvalue $\;1\;$ . Complete the answer by using what you already did.