The signum, aka sign, function is defined for the case where $x\in \mathbb{R}$ as: $$ \text{sign}(x)= \left\{\begin{array}{lr} -1, & \text{for } x< 0\\ 0, & \text{for } x= 0\\ 1, & \text{for } x> 0 \end{array}\right., \qquad(1) $$ which can also be written as
$$\displaystyle\text{sign}(x)=\frac{x}{|x|}, \qquad (2)$$
where $|x|$ is the absolute value of $x$, see wiki here; a definition that holds for the complex case $x\in \mathbb{C}$ by considering $|x|$ as the complex absolute value (modulus) of $x$.
Suppose now that $x\in\mathbb{R}^2$ i.e. $x= [x_2, x_2]^T$, how would you define the sign function?
One possibility would be to write $\displaystyle\text{sign}(x)=[\text{sign}(x_1), \text{sign}(x_2)]^T= [\frac{x_1}{|x_1|}, \frac{x_2}{|x_2|}]^T$, but then what's the usefulness of this?
Wouldn't be more rational to define it, in extension to (2) and particularly the complex-case, as:
$$\displaystyle\text{sign}(x)=\frac{x}{||x||}, \qquad (3)$$
where $||x||$ is a norm (say l2-norm) of $x$, so that $\text{sign}(x)$ would be a unit-norm vector giving the direction of vector $x$?
I'm trying to think about this but I'm not sure if people have already defined this function in the multivariable case. Happy to learn about any references or to exchange on the point of view I presented.
The multivariable signum function is often defined as $(3)$ and is commonly invoked in dynamical systems and control literature (for example, pp. 2 in this paper) where it’s common to have $x \in \mathbb{R}^n$.
The multivariable sign function here, for $x \in \mathbb{R}^n$ is defined as:
$$ \lceil\boldsymbol{x}\rfloor^0= \dfrac{\boldsymbol{x}}{||\boldsymbol{x}||} \tag{1} $$
$$\lceil \boldsymbol{x} \rfloor ^p = ||\boldsymbol{x}||^p \lceil \boldsymbol{x} \rfloor ^0 \tag{2}$$