The convex conic hull of the column space of an $m \times n$ matrix $\mathbf{A}$ is defined as follows
$$\operatorname{ConHull}(\mathbf A) = \{ \mathbf A \mathbf x \mid \mathbf x \geq 0_n \}$$
By this definition, the conical hull of every positive (element-wise positive) matrix should be equal to the non-negative orthant of the $n$-dimensional Euclidean space, right?
It seems very obvious and trivial for me. Is it correct ?
I use the definition of nonnegative orthant as $\{ x \in \mathbb{R}^n | x \geq 0\}$.
For $x \in \mathbb{R}$ and $A \in \mathbb{R}^{n \times 1}$, $A[x]=xA$, for example $$\begin{bmatrix} 1 \\ 2 \end{bmatrix} \begin{bmatrix} 3 \end{bmatrix}=\begin{bmatrix} 1(3) \\ 2(3) \end{bmatrix} = 3\begin{bmatrix} 1 \\ 2 \end{bmatrix}$$
Let $A=\begin{bmatrix} 1 \\ 2\end{bmatrix}$, then the conic hull of $A$ is equal to
$$\{ Ax| x \geq 0 , x \in \mathbb{R}\} = \{ xA| x \geq 0 , x \in \mathbb{R}\} $$
$\{xA| x \geq 0 , x \in \mathbb{R}\} \neq $ non-negative orthant in $\mathbb{R}$. It is just a half-line in $\mathbb{R}^2$.
$\{ x \in \mathbb{R} | x \geq 0\} \neq \{ kA| k \geq 0 , k \in \mathbb{R}\} $
To see that, the set on the left is a subset of $\mathbb{R}$ but the set on the right is a subset of $\mathbb{R}^2$.
Edit: Another counter example:
Let $B=\begin{bmatrix} 1 & 1 \\ 2 & 2\end{bmatrix}$,
then the conic hull of $B$ is equal to
\begin{align}\left\{ y \left| y = B \begin{bmatrix} \lambda \\ \mu\end{bmatrix}, \lambda \ge 0, \mu \ge 0\right.\right\}&= \left\{ y \left| y =\begin{bmatrix} A & A \end{bmatrix} \begin{bmatrix} \lambda \\ \mu\end{bmatrix}, \lambda \ge 0, \mu \ge 0\right.\right\} \\ &= \left\{ y \left| y =\lambda A + \mu A, \lambda \ge 0, \mu \ge 0\right.\right\} \\ &= \left\{ y \left| y =(\lambda + \mu )A, \lambda \ge 0, \mu \ge 0\right.\right\} \\ &=\left\{ y \left| y =kA, k \ge 0, k \in \mathbb{R}\right.\right\} \\\end{align}
which is not equal to the non-negative orthant in $\mathbb{R}^2$. It is just a half-line in $\mathbb{R}^2$.
$\{ x \in \mathbb{R}^2 | x \geq 0\} \neq \{ kA| k \geq 0 , k \in \mathbb{R}\} $
The set on the left is a quadrant but the set on the right is just a half line.