Editing Sign function (section)

== Generalizations ==

=== Complex signum ===
<!-- [[Complex sign function]] and [[Complex signum function]] redirect here -->
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 ===
<!-- Was "Generalization to 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 ===
<!-- Was "Generalized signum function" -->

At real values of <math>x</math>, it is possible to define a [[generalized function]]&ndash;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>.)