Sign function


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 .[1]

Signum function

Definition edit

The signum function of a real number   is a piecewise function which is defined as follows:[1]


Properties edit

The sign function is not continuous at  .

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


It follows that whenever   is not equal to 0 we have


Similarly, for any real number  ,

We can also ascertain that:
The signum function is the derivative of the absolute value function, up to (but not including) the indeterminacy at zero. More formally, in integration theory it is a weak derivative, and in convex function theory the subdifferential of the absolute value at 0 is the interval  , "filling in" the sign function (the subdifferential of the absolute value is not single-valued at 0). Note, the resultant power of   is 0, similar to the ordinary derivative of  . The numbers cancel and all we are left with is the sign of  .

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]

where   is the Heaviside step function using the standard   formalism. Using this identity, it is easy to derive the distributional derivative:[3]

The Fourier transform of the signum function is[4]

where   means taking the Cauchy principal value.

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


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

The signum function has a very simple definition if   is accepted to be equal to 1. Then signum can be written for all real numbers as
The signum function coincides with the limits
as well as,

Here,   is the Hyperbolic tangent and the superscript of -1, above it, is shorthand notation for the inverse function of the Trigonometric function, tangent.

For  , a smooth approximation of the sign function is

Another approximation is
which gets sharper as  ; note that this is the derivative of  . This is inspired from the fact that the above is exactly equal for all nonzero   if  , and has the advantage of simple generalization to higher-dimensional analogues of the sign function (for example, the partial derivatives of  ).

See Heaviside step function § Analytic approximations.

Complex signum edit

The signum function can be generalized to complex numbers as:

for any complex number   except  . The signum of a given complex number   is the point on the unit circle of the complex plane that is nearest to  . Then, for  ,
where   is the 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  :


Another generalization of the sign function for real and complex expressions is  ,[5] which is defined as:

where   is the real part of   and   is the imaginary part of  .

We then have (for  ):


Generalized signum function edit

At real values of  , it is possible to define a generalized function–version of the signum function,   such that   everywhere, including at the point  , unlike  , for which  . 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]

in addition,   cannot be evaluated at  ; and the special name,   is necessary to distinguish it from the function  . (  is not defined, but  .)

Generalization to matrices edit

Thanks to the Polar decomposition theorem, a matrix   (  and  ) can be decomposed as a product   where   is a unitary matrix and   is a self-adjoint, or Hermitian, positive definite matrix, both in  . If   is invertible then such a decomposition is unique and   plays the role of  's signum. A dual construction is given by the decomposition   where   is unitary, but generally different than  . This leads to each invertible matrix having a unique left-signum   and right-signum  .

In the special case where   and the (invertible) matrix  , which identifies with the (nonzero) complex number  , then the signum matrices satisfy   and identify with the complex signum of  ,  . In this sense, polar decomposition generalizes to matrices the signum-modulus decomposition of complex numbers.

See also edit

Notes edit

  1. ^ a b "Signum function - Maeckes".
  2. ^ Weisstein, Eric W. "Sign". MathWorld.
  3. ^ Weisstein, Eric W. "Heaviside Step Function". MathWorld.
  4. ^ Burrows, B. L.; Colwell, D. J. (1990). "The Fourier transform of the unit step function". International Journal of Mathematical Education in Science and Technology. 21 (4): 629–635. doi:10.1080/0020739900210418.
  5. ^ Maple V documentation. May 21, 1998
  6. ^ Yu.M.Shirokov (1979). "Algebra of one-dimensional generalized functions". Theoretical and Mathematical Physics. 39 (3): 471–477. doi:10.1007/BF01017992. Archived from the original on 2012-12-08.