Consider the block matrix $$\textbf{A} = \left[ \begin{array}{ccc} \textbf{X}_{11} & \textbf{X}_{12}\\ \textbf{X}_{21} & \textbf{D}\\ \end{array} \right]$$ where $\textbf{X}_{11} \in \mathbb{C}^{r \times (r+1)}, \textbf{X}_{12} \in \mathbb{C}^{r \times 3nr}, \textbf{X}_{21} \in \mathbb{C}^{3nr \times (r+1)}$. These blocks are supposed to be dense matrices. We have some sparse structure on the block $\textbf{D}$. More precisely, we have that

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

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 $D \in \mathbb{C}^{n \times n}$.

We have that $\textbf{A} \in \mathbb{C}^{(r+3nr) \times (1+r+3nr)}$ and what I'm trying to do is to obtain some upper bound for the norm of its pseudo inverse. Concretely, I want an upper bound for

$$\|\textbf{A}^\dagger\|$$ 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 the advantage of its sparse structure.

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 I hope 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.