Let a matrix $\mathbf{G}$ be a function of a vector $\mathbf{x}=\begin{bmatrix}x_1 \\ x_2\end{bmatrix}$: $$ \mathbf{G} = \begin{bmatrix}x_1 & 0 \\ 0 & x_2\end{bmatrix}.$$
I believe it is valid to notate $\mathbf{G}$ as $\mathbf{G}(\mathbf{x})$ since $$ \mathbf{G} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \cdot \underbrace{\begin{bmatrix}x_1 & x_2\end{bmatrix}}_{\mathbf{x}^T} \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} \cdot \underbrace{\begin{bmatrix}x_1 & x_2\end{bmatrix}}_{\mathbf{x}^T} \cdot \begin{bmatrix} 0 \\ 1 \end{bmatrix} \cdot \begin{bmatrix} 0 & 1 \end{bmatrix}.$$
Now consider a matrix $\mathbf{H}$, a function of vectors $\mathbf{x}$ and $\mathbf{y}=\begin{bmatrix}y_1 \\ y_2\end{bmatrix}$: $$\mathbf{H} = \begin{bmatrix}x_1y_1 & x_1x_2\frac{\partial y_1}{\partial x_1} \\ \frac{\partial y_1}{\partial x_2} & x_2\frac{\partial y_2}{\partial x_2}\end{bmatrix}.$$
Question 1: Is it valid to notate the matrix $\mathbf{H}$ as
(i) $\mathbf{H}(\mathbf{x},\mathbf{y})$, without any indication of the partial derivatives in it?
(ii) $\mathbf{H}(\mathbf{x})$?
(iii) Would either notation in part (i) or (ii) be valid if $\mathbf{H}$ is clearly defined, as in the equation above?
Question 2: For a more generalized case, where we have $\mathbf{x}=\begin{bmatrix}x_1 & x_2 & \cdots & x_n\end{bmatrix}$, $\mathbf{y}=\begin{bmatrix}y_1 & y_2 & \cdots & y_n\end{bmatrix}$, and numerous cross partial derivative and nonlinear terms in a matrix, what would be a valid notation?