Eigenvalues and eigenvectors of a symmetric matrix

Question

If I have a real $n \times n$ matrix with entries $a_{ij}=\delta_{ij}-p_ip_j$ where $p$ is an $n$-vector, how can I find the eigenvalues and the corresponding eigenvectors of this matrix? I have done the case $n=2$ and found that the eigenvalues are $\lambda=1 $ or $=1-|p|^2$.

Answer 1

Note that this matrix looks a little bit like a projector. Every vector orthogonal to $p_i$ is unchanged, whilst $p_i$ itself is rescaled by $1-|p|^2$. If $|p|=1$ this would be a legitimate projection matrix.

The eigenvectors are hence $p_i$, with eigenvalue $1-|p|^2$, as well as all vectors in the $(n-1)$-dimensional subspace orthogonal to $p_i$, with eigenvalue $1$.