Matrix-valued function: understanding

Question

$A(x)=(a_{i,j}(x))_{i,j}: \mathbb{R}^n\to \mathbb{R}^{m\times k}$ is a matrix-valued function.

From my understanding it is a function that maps a vector to a matrix. How does this happen? Is the 'function' a matrix of which the elements are functions? If so, are these just $\mathbb{R}^n-\mathbb{R}$-functions?

I don't really understand the form of a matrix-valued function and how it 'takes' the required input. Maybe an example could be helpful?

Thanks a lot.

Answer 1

Consider first the definition of function in the most general sense. A function $f: A\to B$ between two sets $A$ and $B$ is a process that associates to each element of $A$ a single element of $B$. Following this definition, there is nothing special to take $A=\Bbb R^n$ and $B=\Bbb R^{m\times k}$.

The most trivial function between the above set is the constant function namely, a process that associates to each vector of $\Bbb R^{n}$ the same matrix $M$. But you can also define functions that associate to each vector $v$ of $\Bbb R^n$ a matrix whose entries depend on the coordinates of the $v$ (I am thinking $\Bbb R^n$ with the standard basis). For instance, @MachineLearner gave you a very explicit example. As a function $g:\Bbb R^n \to \Bbb R^m$ can be seen as $$g(x_1,\dots,x_n)=\big(g_1(x_1,\dots,x_n),\dots,g_m(x_1,\dots,x_n)\big)$$

any function $f:\Bbb R^n\to \Bbb R^{m\times k}$ can be seen as $$ f(x_1,\dots,x_n)= \begin{pmatrix} a_{11}(x_1,\dots,x_n) &\cdots & a_{1k}(x_1,\dots,x_n)\\ \dots & \dots & \dots\\ a_{m1}(x_1,\dots,x_n) & \dots & a_{mk}(x_1,\dots,x_n)\end{pmatrix} $$

Hence yes, such a function can be seen as a matrix whose entries are $\Bbb R^n\to\Bbb R$ functions.

Furthermore, such a function appears very often. For instance, consider a smooth function $h:\Bbb R^n\to \Bbb R^m$. The Jacobian $J(h)(x_1,\dots,x_n)$ is a smooth function whose source is again $\Bbb R^n$, but the target is the space of $n\times m$ matrices. Each entry is a $\Bbb R^n\to \Bbb R$ function namely, $${J(h)}_{ij}(x_1,\dots,x_n)=\frac{\partial h_i}{\partial x_j}(x_1,\dots,x_n).$$

Answer 2

As an example: Imagine $x=[x_1,x_2]^T\in \mathbb{R}^2$ Then the function

$$A(x)= \begin{bmatrix} x_1x_2 & x_1^2\sin x_2 & x_2 \\ x_1x_2^3 & x_2\exp(x_1) & 2x_2x_1\sin(x_1) \\ \end{bmatrix}$$

is a map from $\mathbb{R}^2\to \mathbb{R}^{2\times3}$. The entries of the matrix are multivariate scalar functions.

Answer 3

A more geometric example: let $T$ be a function that maps a vector $v$ to a rotation around the vector $v$ by $\frac{\pi}{6}$. Then you can represent this function as a function from $\mathbb{R}^n$ to $\mathbb{R}^{n \times n}$.