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Let $\circ$ be the Hadamard product and consider two matrices $C \in\{0,1\}^{N \times n}$ and $W\in \mathbb{R}^{N\times n}$: $$ C:=\left[\begin{array}{cccc} c_1^1 & c_2^1 & \cdots & c_n^1 \\ \vdots & c_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ c_1^N & \cdots & \cdots & c_n^N \end{array}\right],W:=\left[\begin{array}{cccc} w_1^1 & w_2^1 & \cdots & w_n^1 \\ \vdots & w_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ w_1^N & \cdots & \cdots & w_n^N \end{array}\right]. $$ Therefore, we have: $$ C\circ W:=\left[\begin{array}{cccc} c_1^1 w_1^1 & c_2^1 w_2^1 & \cdots & c_n^1 w_n^1 \\ \vdots & c_2^2 w_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ c_1^N w_1^N & \cdots & \cdots & c_n^N w_n^N \end{array}\right]. $$ We know that $N<n$, $W$ has full rank $N$ and the entries of $C$ can assume values only $0$ or $1$.

Moreover, $C$ has at least one element that is $1$ in every row and every row is different from the other ones. Finally, $C\circ W$ has only non-negative entries, that is if $c_j^i=1$, then $w^i_j>0$ and vice versa.

These assumptions are enough for saying that $C\circ W$ has full rank $N$? If not, what are the right assumptions that one should add for obtaining that rank of $C\circ W$ is $N$?

Let $\circ$ be the Hadamard product and consider two matrices $C \in\{0,1\}^{N \times n}$ and $W\in \mathbb{R}^{N\times n}$: $$ C:=\left[\begin{array}{cccc} c_1^1 & c_2^1 & \cdots & c_n^1 \\ \vdots & c_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ c_1^N & \cdots & \cdots & c_n^N \end{array}\right],W:=\left[\begin{array}{cccc} w_1^1 & w_2^1 & \cdots & w_n^1 \\ \vdots & w_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ w_1^N & \cdots & \cdots & w_n^N \end{array}\right]. $$ Therefore, we have: $$ C\circ W:=\left[\begin{array}{cccc} c_1^1 w_1^1 & c_2^1 w_2^1 & \cdots & c_n^1 w_n^1 \\ \vdots & c_2^2 w_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ c_1^N w_1^N & \cdots & \cdots & c_n^N w_n^N \end{array}\right]. $$ We know that $N<n$, $W$ has full rank $N$ and the entries of $C$ can assume values only $0$ or $1$.

Moreover, $C$ has at least one element that is $1$ in every row and every row is different from the other ones. Finally, $c_j^i=1\Leftrightarrow w^j_i>0$. Therefore, $C\circ W$ has only non-negative entries.

These assumptions are enough for saying that $C\circ W$ has full rank $N$? If not, what are the right assumptions that one should add for obtaining that rank of $C\circ W$ is $N$?

Let $\circ$ be the Hadamard product and consider two matrices $C \in\{0,1\}^{N \times n}$ and $W\in \mathbb{R}^{N\times n}$: $$ C:=\left[\begin{array}{cccc} c_1^1 & c_2^1 & \cdots & c_n^1 \\ \vdots & c_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ c_1^N & \cdots & \cdots & c_n^N \end{array}\right],W:=\left[\begin{array}{cccc} w_1^1 & w_2^1 & \cdots & w_n^1 \\ \vdots & w_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ w_1^N & \cdots & \cdots & w_n^N \end{array}\right]. $$ Therefore, we have: $$ C\circ W:=\left[\begin{array}{cccc} c_1^1 w_1^1 & c_2^1 w_2^1 & \cdots & c_n^1 w_n^1 \\ \vdots & c_2^2 w_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ c_1^N w_1^N & \cdots & \cdots & c_n^N w_n^N \end{array}\right]. $$ We know that $N<n$, $W$ has full rank $N$ and the entries of $C$ can assume values only $0$ or $1$.

Moreover, $C$ has at least one element that is $1$ in every row and every row is different from the other ones. Finally, $C\circ W$ has only non-negative entries, that is if $c_j^i=1$, then $w^i_j>0$.

These assumptions are enough for saying that $C\circ W$ has full rank $N$? If not, what are the right assumptions that one should add for obtaining that rank of $C\circ W$ is $N$?

Let $\circ$ be the Hadamard product and consider two matrices $C \in\{0,1\}^{N \times n}$ and $W\in \mathbb{R}^{N\times n}$: $$ C:=\left[\begin{array}{cccc} c_1^1 & c_2^1 & \cdots & c_n^1 \\ \vdots & c_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ c_1^N & \cdots & \cdots & c_n^N \end{array}\right],W:=\left[\begin{array}{cccc} w_1^1 & w_2^1 & \cdots & w_n^1 \\ \vdots & w_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ w_1^N & \cdots & \cdots & w_n^N \end{array}\right]. $$ Therefore, we have: $$ C\circ W:=\left[\begin{array}{cccc} c_1^1 w_1^1 & c_2^1 w_2^1 & \cdots & c_n^1 w_n^1 \\ \vdots & c_2^2 w_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ c_1^N w_1^N & \cdots & \cdots & c_n^N w_n^N \end{array}\right]. $$ We know that $N<n$, $W$ has full rank $N$ and the entries of $C$ can assume values only $0$ or $1$.

Moreover, $C$ has at least one element that is $1$ in every row and every row is different from the other ones. Finally, $C\circ W$ has only non-negative entries, that is if $c_j^i=1$, then $w^i_j>0$ and vice versa.

These assumptions are enough for saying that $C\circ W$ has full rank $N$? If not, what are the right assumptions that one should add for obtaining that rank of $C\circ W$ is $N$?

Let $\circ$ be the Hadamard product and consider two matrices $C \in\{0,1\}^{N \times n}$ and $W\in \mathbb{R}^{N\times n}$: $$ C:=\left[\begin{array}{cccc} c_1^1 & c_2^1 & \cdots & c_n^1 \\ \vdots & c_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ c_1^N & \cdots & \cdots & c_n^N \end{array}\right],W:=\left[\begin{array}{cccc} w_1^1 & w_2^1 & \cdots & w_n^1 \\ \vdots & w_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ w_1^N & \cdots & \cdots & w_n^N \end{array}\right]. $$ Therefore, we have: $$ C\circ W:=\left[\begin{array}{cccc} c_1^1 w_1^1 & c_2^1 w_2^1 & \cdots & c_n^1 w_n^1 \\ \vdots & c_2^2 w_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ c_1^N w_1^N & \cdots & \cdots & c_n^N w_n^N \end{array}\right]. $$ We know that $N<n$, $W$ has full rank $N$ and the entries of $C$ can assume values only $0$ or $1$. Moreover, $C$ has at least one element that is $1$ in every row and every row is different from the other ones. Finally, $C\circ W$ has only non-negative entries, that is if $c_j^i=1$, then $w^i_j>0$.

These assumptions are enough for saying that $C\circ W$ has full rank $N$? If not, what are the right assumptions that one should add for obtaining that rank of $C\circ W$ is $N$?

Let $\circ$ be the Hadamard product and consider two matrices $C \in\{0,1\}^{N \times n}$ and $W\in \mathbb{R}^{N\times n}$: $$ C:=\left[\begin{array}{cccc} c_1^1 & c_2^1 & \cdots & c_n^1 \\ \vdots & c_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ c_1^N & \cdots & \cdots & c_n^N \end{array}\right],W:=\left[\begin{array}{cccc} w_1^1 & w_2^1 & \cdots & w_n^1 \\ \vdots & w_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ w_1^N & \cdots & \cdots & w_n^N \end{array}\right]. $$ Therefore, we have: $$ C\circ W:=\left[\begin{array}{cccc} c_1^1 w_1^1 & c_2^1 w_2^1 & \cdots & c_n^1 w_n^1 \\ \vdots & c_2^2 w_2^2 & \cdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ c_1^N w_1^N & \cdots & \cdots & c_n^N w_n^N \end{array}\right]. $$ We know that $N<n$, $W$ has full rank $N$ and the entries of $C$ can assume values only $0$ or $1$. 

Moreover, $C$ has at least one element that is $1$ in every row and every row is different from the other ones. Finally, $C\circ W$ has only non-negative entries, that is if $c_j^i=1$, then $w^i_j>0$.

These assumptions are enough for saying that $C\circ W$ has full rank $N$? If not, what are the right assumptions that one should add for obtaining that rank of $C\circ W$ is $N$?