Suppose a square, real and symmetric matrix $G\in\mathbb{R}^{n\times n}$ is given, and it is known to have one zero eigenvalue associated with all ones eigenvector, $1_n$. I'm aware that the (possibly) negative spectrum could be shifted to non-negative, $$(G + c\cdot I)u = Gu + c\cdot Iu = \lambda\cdot u + c\cdot u = (\lambda + c)\cdot u,$$ where $c$ is at least the absolute value of the smallest negative eigenvalues.
a) with the above shifting, are the zero eigenvalue affected (I'm asking because of certain contradictory I found in the literature)?
b) given $J_n=I_n-\frac{1}{n}1_n1_n^T$, what can one say about the spectral decomposition of $$G-c\cdot J_n$$ Is this the same as the above method of shift along the diagonal? I read that that all the eigenvalues are shifted, except for the zero eigenvalue that remains (corresponding to $1_n$), but I wonder how this could be shown.