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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]}

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

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 ,

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]}

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

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:

For , a smooth approximation of the sign function is

The signum function can be generalized to complex numbers as:

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:

We then have (for ):

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]}

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.

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^{a}^{b}"Signum function - Maeckes".*www.maeckes.nl*. **^**Weisstein, Eric W. "Sign".*MathWorld*.**^**Weisstein, Eric W. "Heaviside Step Function".*MathWorld*.**^**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.**^**Maple V documentation. May 21, 1998**^**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.