4
$\begingroup$

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 $\mathbf B:\mathbb R^{n} \rightarrow \mathbb R^{m \times n}$?

$\endgroup$
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.