The signum function of a real number is a piecewise function which is defined as follows:
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 
where is the Heaviside step function using the standard formalism.
Using this identity, it is easy to derive the distributional derivative:
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 ).
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 , which is defined as:
where is the real part of and is the imaginary part of .
We then have (for ):
Generalized signum functionedit
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
in addition, cannot be evaluated at ; and the special name, is necessary to distinguish it from the function . ( is not defined, but .)
Generalization to matricesedit
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.
^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.