Chapter 4.5, Problem 26E

Show that any nonzero vector of the form $au+bv$ is an eigenvector corresponding to $\lambda =1$ for the matrix $P$ in Exercise 25.

25. Suppose that $u$ and $v$ are vectors in $R^n$ such that $u^{T}u=1$, $v^{T}v=1$, and $u^{T}v=0$. Show that $P=u{u}^T+v{v}^T$ is idempotent.