Proper function notation for matrix functions?

Question

For a vector-valued function we have the notation $f:\mathbb R^n\rightarrow \mathbb R^m$.

Is this also a proper notation for a matrix function? Are there any conventions?

For a matrix of one variable, $t$, \begin{align} \mathbf A(t)= \begin{bmatrix} a_{11}(t) & a_{12}(t) & \cdots & a_{1n}(t)\\ a_{12}(t) & a_{22}(t) & \cdots & a_{2n}(t)\\ \vdots &\vdots & \ddots & \vdots\\ a_{n1}(t) & a_{n2}(t) &\cdots & a_{mn}(t) \end{bmatrix}, \end{align} is it correct to write $\mathbf A:\mathbb R \rightarrow \mathbb R^{m\times n}$?

And for a matrix of several variables $\mathbf x=(x_1, \dots, x_n)$, \begin{align} \mathbf B(\mathbf x)= \begin{bmatrix} b_{11}(\mathbf x) & b_{12}(\mathbf x) & \cdots & b_{1n}(\mathbf x)\\ b_{12}(\mathbf x) & b_{22}(\mathbf x) & \cdots & b_{2n}(\mathbf x)\\ \vdots &\vdots & \ddots & \vdots\\ b_{n1}(\mathbf x) & b_{n2}(\mathbf x) &\cdots & b_{mn}(\mathbf x) \end{bmatrix}, \end{align} is the proper notation $\mathbf B:\mathbb R^{n} \rightarrow \mathbb R^{m \times n}$?

Answer 1

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.