In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that returns the sign of a real number. In mathematical notation the sign function is often represented as ${\displaystyle \operatorname {sgn}(x)}$.^[1]

Definition

The signum function of a real number ${\displaystyle x}$ is a piecewise function which is defined as follows:^[1]

$${\displaystyle \operatorname {sgn} x:={\begin{cases}-1&{\text{if ))x<0,\\0&{\text{if ))x=0,\\1&{\text{if ))x>0.\end{cases))}$$

Properties

Any real number can be expressed as the product of its absolute value and its sign function:

$${\displaystyle x=|x|\operatorname {sgn} x.}$$

It follows that whenever ${\displaystyle x}$ is not equal to 0 we have

$${\displaystyle \operatorname {sgn} x={\frac {x}{|x|))={\frac {|x|}{x))\,.}$$

Similarly, for any real number ${\displaystyle x}$,

$${\displaystyle |x|=x\operatorname {sgn} x.}$$

$${\displaystyle \operatorname {sgn} x^{n}=(\operatorname {sgn} x)^{n}.}$$

${\displaystyle [-1,1]}$

${\displaystyle x}$

${\displaystyle x}$

${\displaystyle x}$

$${\displaystyle {\frac ((\text{d))|x|}((\text{d))x))=\operatorname {sgn} x{\text{ for ))x\neq 0\,.}$$

The signum function is differentiable with derivative 0 everywhere except at 0. It is not differentiable at 0 in the ordinary sense, but under the generalised notion of differentiation in distribution theory, the derivative of the signum function is two times the Dirac delta function, which can be demonstrated using the identity ^[2]

$${\displaystyle \operatorname {sgn} x=2H(x)-1\,,}$$

${\displaystyle H(x)}$

${\displaystyle H(0)={\frac {1}{2))}$

$${\displaystyle {\frac ((\text{d))\operatorname {sgn} x}((\text{d))x))=2{\frac ((\text{d))H(x)}((\text{d))x))=2\delta (x)\,.}$$

The Fourier transform of the signum function is^[4]

$${\displaystyle \int _{-\infty }^{\infty }(\operatorname {sgn} x)e^{-ikx}{\text{d))x=PV{\frac {2}{ik)),}$$

${\displaystyle PV}$

The signum can also be written using the Iverson bracket notation:

$${\displaystyle \operatorname {sgn} x=-[x<0]+[x>0]\,.}$$

The signum can also be written using the floor and the absolute value functions:

$${\displaystyle \operatorname {sgn} x={\Biggl \lfloor }{\frac {x}{|x|+1)){\Biggr \rfloor }-{\Biggl \lfloor }{\frac {-x}{|x|+1)){\Biggr \rfloor }\,.}$$

${\displaystyle 0^{0))$

$${\displaystyle \operatorname {sgn} x=0^{\left(-x+\left\vert x\right\vert \right)}-0^{\left(x+\left\vert x\right\vert \right)}\,.}$$

$${\displaystyle \operatorname {sgn} x=\lim _{n\to \infty }{\frac {1-2^{-nx)){1+2^{-nx))}\,.}$$

$${\displaystyle \operatorname {sgn} x=\lim _{n\to \infty }{\frac {2}{\pi )){\rm {arctan))(nx)\,=\lim _{n\to \infty }{\frac {2}{\pi ))\tan ^{-1}(nx)\,.}$$

$${\displaystyle \operatorname {sgn} x=\lim _{n\to \infty }\tanh(nx)\,.}$$

${\displaystyle \tanh(x)}$

For ${\displaystyle k>1}$, a smooth approximation of the sign function is

$${\displaystyle \operatorname {sgn} x\approx \tanh kx\,.}$$

$${\displaystyle \operatorname {sgn} x\approx {\frac {x}{\sqrt {x^{2}+\varepsilon ^{2))))\,.}$$

${\displaystyle \varepsilon \to 0}$

${\displaystyle {\sqrt {x^{2}+\varepsilon ^{2))))$

${\displaystyle x}$

${\displaystyle \varepsilon =0}$

${\displaystyle {\sqrt {x^{2}+y^{2))))$

See Heaviside step function § Analytic approximations.

Complex signum

The signum function can be generalized to complex numbers as:

$${\displaystyle \operatorname {sgn} z={\frac {z}{|z|))}$$

${\displaystyle z}$

${\displaystyle z=0}$

${\displaystyle z}$

${\displaystyle z}$

${\displaystyle z\neq 0}$

$${\displaystyle \operatorname {sgn} z=e^{i\arg z}\,,}$$

${\displaystyle \arg }$

For reasons of symmetry, and to keep this a proper generalization of the signum function on the reals, also in the complex domain one usually defines, for ${\displaystyle z=0}$:

$${\displaystyle \operatorname {sgn}(0+0i)=0}$$

Another generalization of the sign function for real and complex expressions is ${\displaystyle {\text{csgn))}$,^[5] which is defined as:

$${\displaystyle \operatorname {csgn} z={\begin{cases}1&{\text{if ))\mathrm {Re} (z)>0,\\-1&{\text{if ))\mathrm {Re} (z)<0,\\\operatorname {sgn} \mathrm {Im} (z)&{\text{if ))\mathrm {Re} (z)=0\end{cases))}$$

${\displaystyle {\text{Re))(z)}$

${\displaystyle z}$

${\displaystyle {\text{Im))(z)}$

${\displaystyle z}$

We then have (for ${\displaystyle z\neq 0}$):

$${\displaystyle \operatorname {csgn} z={\frac {z}{\sqrt {z^{2))))={\frac {\sqrt {z^{2))}{z)).}$$

Generalized signum function

At real values of ${\displaystyle x}$, it is possible to define a generalized function–version of the signum function, ${\displaystyle \varepsilon (x)}$ such that ${\displaystyle \varepsilon (x)^{2}=1}$ everywhere, including at the point ${\displaystyle x=0}$, unlike ${\displaystyle \operatorname {sgn} }$, for which ${\displaystyle (\operatorname {sgn} 0)^{2}=0}$. This generalized signum allows construction of the algebra of generalized functions, but the price of such generalization is the loss of commutativity. In particular, the generalized signum anticommutes with the Dirac delta function^[6]

$${\displaystyle \varepsilon (x)\delta (x)+\delta (x)\varepsilon (x)=0\,;}$$

${\displaystyle \varepsilon (x)}$

${\displaystyle x=0}$

${\displaystyle \varepsilon }$

${\displaystyle \operatorname {sgn} }$

${\displaystyle \varepsilon (0)}$

${\displaystyle \operatorname {sgn} 0=0}$

Generalization to matrices

Thanks to the Polar decomposition theorem, a matrix ${\displaystyle {\boldsymbol {A))\in \mathbb {K} ^{n\times n))$ (${\displaystyle n\in \mathbb {N} }$ and ${\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \))$) can be decomposed as a product ${\displaystyle {\boldsymbol {Q)){\boldsymbol {P))}$ where ${\displaystyle {\boldsymbol {Q))}$ is a unitary matrix and ${\displaystyle {\boldsymbol {P))}$ is a self-adjoint, or Hermitian, positive definite matrix, both in ${\displaystyle \mathbb {K} ^{n\times n))$. If ${\displaystyle {\boldsymbol {A))}$ is invertible then such a decomposition is unique and ${\displaystyle {\boldsymbol {Q))}$ plays the role of ${\displaystyle {\boldsymbol {A))}$'s signum. A dual construction is given by the decomposition ${\displaystyle {\boldsymbol {A))={\boldsymbol {S)){\boldsymbol {R))}$ where ${\displaystyle {\boldsymbol {R))}$ is unitary, but generally different than ${\displaystyle {\boldsymbol {Q))}$. This leads to each invertible matrix having a unique left-signum ${\displaystyle {\boldsymbol {Q))}$ and right-signum ${\displaystyle {\boldsymbol {R))}$.

In the special case where ${\displaystyle \mathbb {K} =\mathbb {R} ,\ n=2,}$ and the (invertible) matrix ${\displaystyle {\boldsymbol {A))=\left[{\begin{array}{rr}a&-b\\b&a\end{array))\right]}$, which identifies with the (nonzero) complex number ${\displaystyle a+\mathrm {i} b=c}$, then the signum matrices satisfy ${\displaystyle {\boldsymbol {Q))={\boldsymbol {P))=\left[{\begin{array}{rr}a&-b\\b&a\end{array))\right]/|c|}$ and identify with the complex signum of ${\displaystyle c}$, ${\displaystyle \operatorname {sgn} c=c/|c|}$. In this sense, polar decomposition generalizes to matrices the signum-modulus decomposition of complex numbers.