Let
$M = J + H = \begin{bmatrix} 0 & \vec{1}^{\intercal}\\ \vec{1} & K + \gamma^{-1}I \end{bmatrix} = \begin{bmatrix} 0 & \vec{1}^{\intercal}\\ \vec{1} & 0 \end{bmatrix}_J + \begin{bmatrix} 0 & 0\\ 0 & K + \gamma^{-1}I \end{bmatrix}_{H} $
where $\vec{1}$ is a vector of $n$ ones, $K$ is a $N \times N$ symmetric matrix, $\gamma > 0$ and $I$ is the identity.
Is there a lower bound on the singular values of M?
I noticed that the singular values of M are close to the ones of H but I cannot demonstrate it.