Let $\mathbf{v}$ be a non-zero (column) vector in $\mathbb{R}^n$.
(a) Find an explicit formula for the matrix $P_\mathbf{v}$ corresponding to the projection of $\mathbb{R}^n$ to the orthogonal complement of the one-dimensional subspace spanned by $\mathbf{v}$.
(b) What are the eigenvalues and eigenvectors of $P_\mathbf{v}$? Compute the dimensions of the associated eigenspaces. Justify your answers.
Wouldn't the matrix be a diagonal matrix such that the eigenvalues are $0$ and $1$ or $-1$? Since it is an orthogonal complement of the subspace spanned by $\mathbf{v}$, wouldn't the matrix also have rows that contain the null space?