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Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M,b]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb R_{\geq0,\leq b}^{n\times n}\}?$$

What is a good upper bound to $$\max_{M\in\Bbb Z_{\geq0,\leq b}^{n\times n},\mathsf{rank}(M)=r}\min_{Q\in\mathscr{D}[M,b]}\mathsf{rank}(Q)?$$

Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M]=\{Q\in\Bbb\{0,1\}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb\{0,1\}^{n\times n}\}?$$

What is a good upper bound to $$\max_{M\in\Bbb\{0,1\}^{n\times n},\mathsf{rank}(M)=r}\min_{Q\in\mathscr{D}[M]}\mathsf{rank}(Q)?$$

Given matrix $M\in\Bbb R_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize matrices $Q\in\Bbb R_{\geq0,\leq b}^{n\times n}$ such that $$\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb R_{\geq0,\leq b}^{n\times n}?$$

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M,b]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb R_{\geq0,\leq b}^{n\times n}\}?$$

What is a good upper bound to $$\max_{M\in\Bbb Z_{\geq0,\leq b}^{n\times n},\mathsf{rank}(M)=r}\min_{Q\in\mathscr{D}[M,b]}\mathsf{rank}(Q)?$$

Given matrix $M\in\Bbb R^{n\times n}$, is there a nice method to characterize matrices $Q\in\Bbb R^{n\times n}$ such that $$\mathsf{rank}(M+Q)\leq s\cdot\mathsf{rank}(Q)$$with some fixed $s>0$?

What if $M\in\Bbb Z_{\geq0}^{n\times n}$ with highest entry $b$?

Given matrix $M\in\Bbb R_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize matrices $Q\in\Bbb R_{\geq0,\leq b}^{n\times n}$ such that $$\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb R_{\geq0,\leq b}^{n\times n}?$$