Let $M\in\{-1,0,+1\}^{n\times n}$ be a matrix of rank $r$.

Consider the matrix $f(M)\in\{0,+1\}^{mn\times mn}$ where $0$ in $M$ is replaced by $m\times m$ all $0$ matrix, $+1$ in $M$ is replaced by $m\times m$ identity matrix and $-1$ in $M$ is replaced by $m\times m$ anti-identity matrix (anti-diagonal matrix having $1$s on anti-diagonal whose square is identity).

Example at $m=2$:

$0$ is replaced by $\begin{bmatrix}0&0\\0&0\end{bmatrix}$, $+1$ is replaced by $\begin{bmatrix}1&0\\0&1\end{bmatrix}$ and $-1$ is replaced by $\begin{bmatrix}0&1\\1&0\end{bmatrix}$.

1. Is the real rank of the new matrix $O(r)$? Is there a precise bound?

2. Is the $\mathbb F_2$ rank of the new matrix $O(r)$? Is there a precise bound?

Let $M\in\{-1,0,+1\}^{n\times n}$ be a matrix of rank $r$.

Consider the matrix $f(M)\in\{0,+1\}^{mn\times mn}$ where $0$ in $M$ is replaced by $m\times m$ all $0$ matrix, $+1$ in $M$ is replaced by $m\times m$ identity matrix and $-1$ in $M$ is replaced by $m\times m$ anti-identity matrix (anti-diagonal matrix having $1$s on anti-diagonal whose square is identity).

Example at $m=2$:

$0$ is replaced by $\begin{bmatrix}0&0\\0&0\end{bmatrix}$, $+1$ is replaced by $\begin{bmatrix}1&0\\0&1\end{bmatrix}$ and $-1$ is replaced by $\begin{bmatrix}0&1\\1&0\end{bmatrix}$.

1. Is the real rank of the new matrix $O(r)$? Is there a precise bound?

2. Is the $\mathbb F_2$ rank of the new matrix $O(r)$? Is there a precise bound?

On the real and finite field rank of a $0/1$ matrix - II