Given a differentiable vector field $\mathrm v : \mathbb R^n \to \mathbb R^n$ and a matrix $\mathrm A \in \mathbb R^{n \times n}$, let function $f : \mathbb R^n \to \mathbb R$ be defined by
$$f (\mathrm x) := \langle \mathrm v (\mathrm x), \mathrm A \mathrm v (\mathrm x) \rangle$$
whose directional derivative in the direction of $\mathrm y \in \mathbb R^n$ at $\mathrm x \in \mathbb R^n$ is
$$D_{\mathrm y} f (\mathrm x) := \lim_{h \to 0} \frac{f (\mathrm x + h \mathrm y) - f (\mathrm x)}{h} = \cdots = \langle \mathrm y, \mathrm J_{\mathrm v}^\top (\mathrm x) \, \mathrm A \, \mathrm v (\mathrm x) \rangle + \langle \mathrm J_{\mathrm v}^\top (\mathrm x) \, \mathrm A^\top \mathrm v (\mathrm x) , \mathrm y \rangle$$
where matrix $\mathrm J_{\mathrm v} (\mathrm x)$ is the Jacobian of vector field $\rm v$ at $\mathrm x \in \mathbb R^n$. Thus, the gradient of $f$ is
$$\nabla_{\mathrm x} f (\mathrm x) = \mathrm J_{\mathrm v}^\top (\mathrm x) \left( \mathrm A + \mathrm A^\top \right) \mathrm v (\mathrm x)$$
The description on the right is very common and serves to give the reader a schematic description of the matrices.
I would write the equation above as follows, although this is my preference:
\begin{equation}
x^\top \Lambda x =
\begin{bmatrix}
x_1 & \cdots & x_n
\end{bmatrix}
\begin{bmatrix}
\lambda_1 & & \\
& \ddots & \\
& & \lambda_n
\end{bmatrix}
\begin{bmatrix}
x_1 \\ \vdots \\ x_n
\end{bmatrix}, \tag{1} \label{myeq}
\end{equation}
where it is understood that the zeros occupy the blank spaces above and below the diagonal.
EDIT: Comments in the sequel address the OP's comments to this answer.
For $n \in \mathbb{N}$, let $\langle n \rangle := \{ 1, \dots, n\}$. If $A:\langle m \rangle \times \langle n \rangle \to \mathbb{F}$ is a function and $\mathbb{F}$ is a set (typically a field), then $A$ is called an $m$-by-$n$ matrix. By convention, the value $A(i,j)$ is abbreviated to $A_{ij}$ or $a_{ij}$.
Given the nature of the function, it is often convenient to represent a matrix via a rectangular array. For instance, the matrix $A=\{a_{11},a_{12},a_{21},a_{22}\}$ is written as
\begin{equation}
A=
\begin{bmatrix}
a_{11} & a_{12 } \\
a_{21} & a_{22}
\end{bmatrix}.
\end{equation}
In general, it is understood that $A=[a_{ij}] \in \textsf{M}_{m \times n}(\mathbb{F})$ is the function $\{a_{ij}\}_{i\in\langle m \rangle, j \in \langle n \rangle}$.
It is customary in linear algebra/matrix theory journals to define a matrix and give its schematic description as in \eqref{myeq}.
Here is an example from a 2017 paper I co-authored with Charles R. Johnson, who is widely considered to be the best matrix theorist in the world:
I have published and refereed in the top journals in matrix theory and linear algebra and I have never objected to or had a referee object to the practice you describe in your post.
Best Answer
It's correct as long as you define $a_{ij}:\mathbb{R}\to\mathbb{R}$ or $b_{ij}:\mathbb{R}^n\to\mathbb{R}$, and you describe how you are identifying the set of $m\times n$ matrices with $\mathbb{R}^{m\times n}$.
You can avoid this if you define $M_{m\times n}$ to be the set of all $m\times n$ matrices with real entries, and set $A:\mathbb{R}\to M_{m\times n}$ and $B:\mathbb{R}^n\to M_{m\times n}$ as above. Then the functions $a_{ij}:\mathbb{R}\to\mathbb{R}$ and $b_{ij}:\mathbb{R}^n\to\mathbb{R}$ are given by $$ a_{ij}(t) = (A(t))_{ij} \quad\text{and}\quad b_{ij}(x) = (B(x))_{ij}, $$ where $E_{ij}$ denotes the entry in the $i$th row and $j$th column of $E\in M_{m \times n}$. This is the technique that I've seen used.