Consider the block matrix given by

$$\textbf{D} = \left[ \begin{array}{ccc} \left[ \begin{array}{ccc} D & \ldots & D\\ \vdots & \ddots & \vdots\\ D & \ldots & D\\ \end{array}\right] & \textbf{X} & \textbf{X}\\ \textbf{X} & \left[ \begin{array}{ccc} D & \ldots & D\\ \vdots & \ddots & \vdots\\ D & \ldots & D\\ \end{array}\right] & \textbf{X}\\ \textbf{X} & \textbf{X} & \left[ \begin{array}{ccc} D & \ldots & D\\ \vdots & \ddots & \vdots\\ D & \ldots & D\\ \end{array}\right]\\ \end{array}\right]$$

where each $\textbf{X}$ represents a dense matrix (but possible different from each other) in $\mathbb{C}^{nr \times nr}$. Each matrix in the form

$$\left[ \begin{array}{ccc} D & \ldots & D\\ \vdots & \ddots & \vdots\\ D & \ldots & D\\ \end{array}\right]\\$$ is composed by $r \times r$ of these matrices $D$, which are diagonal matrices (but possible different from each other) in $\mathbb{C}^{n \times n}$. So each of its $r^2$ entries is a diagonal matrix $D \in \mathbb{C}^{n \times n}$.

This matrix has a sparse pattern and is supposed to be invertible. What I'm trying to do is to obtain some upper bound for the norm of its inverse. Concretely, I want an upper bound for

$$\|\textbf{D}^{-1}\|$$ where the norm can be the spectral norm or the Frobenius norm. I am looking for some bound in terms of these blocks, in a way it take advantage of its sparsity.

I already looked through several papers and searched on the internet, but couldn't find anything helpful. All my ideas also didn't work, so my last hope is that someone here may have a good idea. This looks to be a very specific problem, and since I'm not so familiar with results about sparse matrices, the best option is to share with you in the hope someone knows something about it.

Thank you!

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