equalizing diagonals of a matrix inverse

Let $Z$ be a square matrix; denote by $\text{diag}_k(Z)$ the matrix containing the center $2k+1$ diagonals of $Z$; i.e., if $T=\text{diag}_k(Z)$, then $$T_{ij}=Z_{ij}, \;\mathrm{if}\; |i-j|\leq k, \mathrm{and }\ T_{ij}=0 \; \mathrm{otherwise}.$$

Let $G_1$ be a nonnegative Hermitian matrix, and construct $X_1=(G_1+\lambda I)^{-1}$ for some given positive $\lambda$. Let $G_2$ be a Hermitian matrix that satisfies $G_2=\text{diag}_k(G_2)$ for some $k$, and let $X_2=(G_2+I)^{-1}$ be such that $$\text{diag}_k(X_1)=\text{diag}_k(X_2).$$

Two questions:

1. Is there any closed form for $G_2$ for an arbitrary $k$?

2. For an arbitrary $k$, when is the solution $G_2$ a non-negative matrix?

Thanks

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1. $G_2 = X_2^{-1}-I$, but that's probably not what you're looking for :-) 2. You mean, elementwise nonnegative? The question is then: when is $X_2^{-1}-I$ elementwise nonnegative--- since I edited several things in your question, I'll wait for you to further clarify the question before spending more time on it. – Suvrit Oct 27 '13 at 20:23
1) The matrix $X_2$ is not known and is really what one must optimize for so that \$G_2=diag_k(G_2). 2) no, I mean a positive semi-definite matrix. – john stark Oct 28 '13 at 21:17