Let $diag_k(Z)$, where $Z$ is a matrix be a matrix of the same size holding the center $2k+1$ diagonals of $Z$ and zeros elsewhere, i.e. if $T=diag_k(Z$ then $$T_{ij}=Z_{ij}, \;\mathrm{if}\; |i-j|\leq k, \mathrm{and} T_{ij}=0 \; \mathrm{otherwise}.$$

Now, let $G_1$ be any non-negative Hermitian matrix and construct $X_1=(G_1+\lambda I)^{-1}$ for some positive $\lambda$. Let $G_2$ be an Hermitian matrix that satisfies $G_2=diag_k(G_2)$ for some $k$, and define $X_2=(G_1+I)^{-1}$ that satisfies $$diag_k(X_1)=diag_k(X_2).$$

Two questions:

1. Is there any closed form for $G_2$?

2. When is $G_2$ non-negative?

Thanks