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{{Short description|Function returning minus 1, zero or plus 1}}
{{Redirect|Sgn||SGN (disambiguation)}}
{{distinguish|Sign relation|Sine function}}

[[Image:Signum function.svg|thumb|Signum function <math>y = \sgn x</math>]]

In [[mathematics]], the '''sign function''' or '''signum function''' (from ''[[wikt:signum#Latin|signum]]'', [[Latin language|Latin]] for "sign") is a [[Function (mathematics)|function]] that has the value {{math|−1}}, {{math|+1}} or {{math|0}} according to whether the [[Sign (mathematics)|sign]] of a given [[real number]] is positive or negative, or the given number is itself zero. In [[mathematical notation]] the sign function is often represented as <math>\sgn x</math> or <math>\sgn (x)</math>.<ref name=":0">{{Cite web|title=Signum function - Maeckes|url=http://www.maeckes.nl/Signum%20functie%20GB.html|access-date=|website=www.maeckes.nl}}</ref>

==Definition==
The signum function of a real number <math>x</math> is a [[piecewise]] function which is defined as follows:<ref name=":0" />
<math display="block"> \sgn x :=\begin{cases}
-1 & \text{if } x < 0, \\
0 & \text{if } x = 0, \\
1 & \text{if } x > 0. \end{cases}</math>

The [[law of trichotomy]] states that every real number must be positive, negative or zero.
The signum function denotes which unique category a number falls into by mapping it to one of the values {{math|−1}}, {{math|+1}} or {{math|0,}} which can then be used in mathematical expressions or further calculations.

For example:
<math display="block">\begin{array}{lcr}
\sgn(2) &=& +1\,, \\
\sgn(\pi) &=& +1\,, \\
\sgn(-8) &=& -1\,, \\
\sgn(-\frac{1}{2}) &=& -1\,, \\
\sgn(0) &=& 0\,.
\end{array}</math>

== Basic properties ==
Any real number can be expressed as the product of its [[absolute value]] and its sign:
<math display="block"> x = |x| \sgn x\,.</math>

It follows that whenever <math>x</math> is not equal to 0 we have
<math display="block"> \sgn x = \frac{x}{|x|} = \frac{|x|}{x}\,.</math>

Similarly, for ''any'' real number <math>x</math>,
<math display="block"> |x| = x\sgn x\,. </math>
We can also be certain that:
<math display="block">\sgn (xy)=(\sgn x)(\sgn y)\,,</math>
and so
<math display="block">\sgn (x^n)=(\sgn x)^n\,.</math>

== Some algebraic identities ==
The signum can also be written using the [[Iverson bracket]] notation:
<math display="block">\sgn x = -[x < 0] + [x > 0] \,.</math>

The signum can also be written using the [[Floor and ceiling functions|floor]] and the absolute value functions:
<math display="block">\sgn x = \Biggl\lfloor \frac{x}{|x|+1} \Biggr\rfloor -
\Biggl\lfloor \frac{-x}{|x|+1} \Biggr\rfloor \,.</math>
If <math>0^0</math> is accepted to be equal to 1, the signum can also be written for all real numbers as
<math display="block">\sgn x = 0^ \left ( - x + \left\vert x \right\vert \right ) - 0^ \left ( x + \left\vert x \right\vert \right ) \,.</math>

== Properties in mathematical analysis ==

=== Discontinuity at zero ===
[[File:Discontinuity of the sign function at 0.svg|thumb|300px|The sign function is not [[continuous function | continuous]] at <math>x=0</math>.]]

Although the sign function takes the value {{math|−1}} when <math>x</math> is negative, the ringed point {{math|(0, −1)}} in the plot of <math>\sgn x</math> indicates that this is not the case when <math>x=0</math>. Instead, the value jumps abruptly to the solid point at {{math|(0, 0)}} where <math>\sgn(0)=0</math>. There is then a similar jump to <math>\sgn(x)=+1</math> when <math>x</math> is positive. Either jump demonstrates visually that the sign function <math>\sgn x</math> is discontinuous at zero, even though it is continuous at any point where <math>x</math> is either positive or negative.

These observations are confirmed by any of the various equivalent formal definitions of [[Continuous function|continuity]] in [[mathematical analysis]]. A function <math>f(x)</math>, such as <math>\sgn(x),</math> is continuous at a point <math>x=a</math> if the value <math>f(a)</math> can be approximated arbitrarily closely by the [[sequence]] of values <math>f(a_1),f(a_2),f(a_3),\dots,</math> where the <math>a_n</math> make up any infinite sequence which becomes arbitrarily close to <math>a</math> as <math>n</math> becomes sufficiently large. In the notation of mathematical [[Limit of a sequence|limit]]s, continuity of <math>f</math> at <math>a</math> requires that <math>f(a_n) \to f(a)</math> as <math>n \to \infty</math> for any sequence <math>\left(a_n\right)_{n=1}^\infty</math> for which <math>a_n \to a.</math> The arrow symbol can be read to mean ''approaches'', or ''tends to'', and it applies to the sequence as a whole.

This criterion fails for the sign function at <math>a=0</math>. For example, we can choose <math>a_n</math> to be the sequence <math>1,\tfrac{1}{2},\tfrac{1}{3},\tfrac{1}{4},\dots,</math> which tends towards zero as <math>n</math> increases towards infinity. In this case, <math>a_n \to a</math> as required, but <math>\sgn(a)=0</math> and <math>\sgn(a_n)=+1</math> for each <math>n,</math> so that <math>\sgn(a_n) \to 1 \neq \sgn(a)</math>. This counterexample confirms more formally the discontinuity of <math>\sgn x</math> at zero that is visible in the plot.

Despite the sign function having a very simple form, the step change at zero causes difficulties for traditional [[calculus]] techniques, which are quite stringent in their requirements. Continuity is a frequent constraint. One solution can be to approximate the sign function by a smooth continuous function; others might involve less stringent approaches that build on classical methods to accommodate larger classes of function.

=== Smooth approximations and limits ===
The signum function can be given as a number of different (pointwise) limits:
<math display="block">\begin{align}
\sgn x &= \lim_{n\to\infty}\frac{1-2^{-nx}}{1+2^{-nx}}\\
&= \lim_{n\to\infty}\frac{2}{\pi}\operatorname{arctan}(nx)\\
&= \lim_{n\to\infty}\tanh(nx)\\
&= \lim_{\varepsilon\to 0} \frac{x}{\sqrt{x^2 + \varepsilon^2}}.
\end{align}</math>
Here, <math>\tanh</math> is the [[hyperbolic tangent]], and <math>\operatorname{arctan}</math> is the [[arctan|inverse tangent]]. The last of these is the derivative of <math>\sqrt{x^2+\varepsilon ^2}</math>. This is inspired from the fact that the above is exactly equal for all nonzero <math>x</math> if <math>\varepsilon=0</math>, and has the advantage of simple generalization to higher-dimensional analogues of the sign function (for example, the partial derivatives of <math>\sqrt{x^2+y^2}</math>).

See ''{{section link|Heaviside step function#Analytic approximations}}''.

=== Differentiation ===

The signum function <math>\sgn x</math> is [[Differentiable function|differentiable]] everywhere except when <math>x=0.</math> Its [[derivative]] is zero when <math>x</math> is non-zero:
<math display="block"> \frac{\text{d}\, (\sgn x)}{\text{d}x} = 0 \qquad \text{for } x \ne 0\,.</math>

This follows from the differentiability of any [[constant function]], for which the derivative is always zero on its domain of definition. The signum <math>\sgn x</math> acts as a constant function when it is restricted to the negative [[Interval (mathematics)#Definitions and terminology|open region]] <math>x<0,</math> where it equals {{math|−1}}. It can similarly be regarded as a constant function within the positive open region <math>x>0,</math> where the corresponding constant is {{math|+1}}. Although these are two different constant functions, their derivative is equal to zero in each case.

It is not possible to define a classical derivative at <math>x=0</math>, because there is a discontinuity there.

Although it is not differentiable at <math>x=0</math> in the ordinary sense, under the generalized notion of differentiation in [[distribution (mathematics)|distribution theory]],
the derivative of the signum function is two times the [[Dirac delta function]]. This can be demonstrated using the identity <ref>{{MathWorld |title=Sign |id=Sign}}</ref>
<math display="block"> \sgn x = 2 H(x) - 1 \,,</math>
where <math>H(x)</math> is the [[Heaviside step function]] using the standard <math>H(0)=\frac{1}{2}</math> formalism.
Using this identity, it is easy to derive the distributional derivative:<ref>{{MathWorld |title=Heaviside Step Function |id=HeavisideStepFunction}}</ref>
<math display="block"> \frac{\text{d}\sgn x}{\text{d}x} = 2 \frac{\text{d} H(x)}{\text{d}x} = 2\delta(x) \,.</math>

=== Integration ===

The signum function has a [[definite integral]] between any pair of finite values {{mvar|a}} and {{mvar|b}}, even when the interval of integration includes zero. The resulting integral for {{mvar|a}} and {{mvar|b}} is then equal to the difference between their absolute values:
<math display="block"> \int_a^b (\sgn x) \, \text{d}x = |b| - |a| \,.</math>

In fact, the signum function is the derivative of the absolute value function, except where there is an abrupt change in [[slope|gradient]] at zero:
<math display="block"> \frac{\text{d} |x|}{\text{d}x} = \sgn x \qquad \text{for } x \ne 0\,.</math>

We can understand this as before by considering the definition of the absolute value <math>|x|</math> on the separate regions <math>x>0</math> and <math>x<0.</math> For example, the absolute value function is identical to <math>x</math> in the region <math>x>0,</math> whose derivative is the constant value {{math|+1}}, which equals the value of <math>\sgn x</math> there.

Because the absolute value is a [[convex function]], there is at least one [[subderivative]] at every point, including at the origin. Everywhere except zero, the resulting [[subdifferential]] consists of a single value, equal to the value of the sign function. In contrast, there are many subderivatives at zero, with just one of them taking the value <math>\sgn(0) = 0</math>. A subderivative value {{math|0}} occurs here because the absolute value function is at a minimum. The full family of valid subderivatives at zero constitutes the subdifferential interval <math>[-1,1]</math>, which might be thought of informally as "filling in" the graph of the sign function with a vertical line through the origin, making it continuous as a two dimensional curve.

In integration theory, the signum function is a [[weak derivative]] of the absolute value function. Weak derivatives are equivalent if they are equal [[almost everywhere]], making them impervious to isolated anomalies at a single point. This includes the change in gradient of the absolute value function at zero, which prohibits there being a classical derivative.

=== Fourier transform ===
The [[Fourier transform]] of the signum function is<ref>{{cite journal|last1=Burrows|first1=B. L.|last2=Colwell|first2=D. J.|title=The Fourier transform of the unit step function|journal=International Journal of Mathematical Education in Science and Technology|date=1990|volume=21|issue=4|pages=629–635|doi=10.1080/0020739900210418}}</ref>
<math display="block">PV\int_{-\infty}^\infty (\sgn x) e^{-ikx}\text{d}x = \frac{2}{ik} \qquad \text{for } k \ne 0,</math>
where <math>PV</math> means taking the [[Cauchy principal value]].

== Generalizations ==

=== Complex signum ===

The signum function can be generalized to [[complex numbers]] as:
<math display="block">\sgn z = \frac{z}{|z|} </math>
for any complex number <math>z</math> except <math>z=0</math>. The signum of a given complex number <math>z</math> is the [[point (geometry)|point]] on the [[unit circle]] of the [[complex plane]] that is nearest to <math>z</math>. Then, for <math>z\ne 0</math>,
<math display="block">\sgn z = e^{i\arg z}\,,</math>
where <math>\arg</math> is the [[Argument (complex analysis)|complex argument function]].

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 <math>z=0</math>:
<math display="block">\sgn(0+0i)=0</math>

Another generalization of the sign function for real and complex expressions is <math>\text{csgn}</math>,<ref>Maple V documentation. May 21, 1998</ref> which is defined as:
<math display="block">
\operatorname{csgn} z= \begin{cases}
1 & \text{if } \mathrm{Re}(z) > 0, \\
-1 & \text{if } \mathrm{Re}(z) < 0, \\
\sgn \mathrm{Im}(z) & \text{if } \mathrm{Re}(z) = 0
\end{cases}
</math>
where <math>\text{Re}(z)</math> is the real part of <math>z</math> and <math>\text{Im}(z)</math> is the imaginary part of <math>z</math>.

We then have (for <math>z\ne 0</math>):
<math display="block">\operatorname{csgn} z = \frac{z}{\sqrt{z^2}} = \frac{\sqrt{z^2}}{z}. </math>

=== Polar decomposition of matrices ===


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

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

=== Signum as a generalized function ===


At real values of <math>x</math>, it is possible to define a [[generalized function]]–version of the signum function, <math>\varepsilon (x)</math> such that <math>\varepsilon (x)^2=1</math> everywhere, including at the point <math>x=0</math>, unlike <math>\sgn</math>, for which <math>(\sgn 0)^2=0</math>. 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<ref name="Algebra">
{{cite journal
|author = Yu.M.Shirokov
|title = Algebra of one-dimensional generalized functions
|journal = [[Theoretical and Mathematical Physics]]
|year = 1979
|volume = 39
|issue = 3
|pages = 471–477
|url = http://springerlink.metapress.com/content/w3010821x8267824/?p=5bb23f98d846495c808e0a2e642b983a&pi=3
|archive-url = https://archive.today/20121208232109/http://springerlink.metapress.com/content/w3010821x8267824/?p=5bb23f98d846495c808e0a2e642b983a&pi=3
|url-status = dead
|archive-date = 2012-12-08
|doi = 10.1007/BF01017992
|bibcode = 1979TMP....39..471S
}}</ref>
<math display="block">\varepsilon (x) \delta(x)+\delta(x) \varepsilon(x) = 0 \, ;</math>
in addition, <math>\varepsilon (x)</math> cannot be evaluated at <math>x=0</math>; and the special name, <math>\varepsilon</math> is necessary to distinguish it from the function <math>\sgn</math>. (<math>\varepsilon (0)</math> is not defined, but <math>\sgn 0=0</math>.)

==See also==
* [[Absolute value]]
* [[Heaviside step function]]
* [[Negative number]]
* [[Rectangular function]]
* [[Sigmoid function]] ([[Hard sigmoid]])
* [[Step function]] ([[Piecewise constant function]])
* [[Three-way comparison]]
* [[Zero crossing]]
* [[Polar decomposition]]

==Notes==
<references/>

{{DEFAULTSORT:Sign Function}}
[[Category:Special functions]]
[[Category:Unary operations]]

Sign function - Revision history

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