Given a symmetric matrix $A$ and a vector $v$, will adding $v v^T$ to $A$ always increase or keep the eigenvalues of $A$ the same? The sum of the eigenvectors of $A + v v^T$ should be greater or equal to the sum of the eigenvectors of $A$ due to $tr(A + v v^T) = tr(A) + tr(v v^T)$, but one could increase one of the eigenvalues of the result matrix while decreasing another and still satisfy the sum increasing.