Sign function

In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that has the value −1, sgn⁡x or x is a piecewise function which is defined as follows: 0. \\end{cases}"}}'>sgn⁡x:={−1if x<0,0if x=0,1if x>0.

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 0, which can then be used in mathematical expressions or further calculations.

For example: x is not equal to 0 we have |x|=xsgn⁡x. We can also be certain that: x|x|×x|x|=x|x|×|x|x =1for x≠0,

the following two statements hold true for any integer (sgn⁡x)2n−1=sgn⁡xfor x≠0.

Hence also:

sgn⁡x=⌊x|x|+1⌋−⌊−x|x|+1⌋. If x=0.

Although the sign function takes the value (0, −1) in the plot of (0, 0) where x is positive. Either jump demonstrates visually that the sign function f(x), such as f(a) can be approximated arbitrarily closely by the sequence of values a as a requires that (an)n=1∞ for which an to be the sequence an→a as required, but n, so that sgn⁡x=limn→∞1−2−nx1+2−nx=limn→∞2πarctan⁡(nx)=limn→∞tanh⁡(nx)=limε→0xx2+ε2. Here, x2+ε2. This is inspired from the fact that the above is exactly equal for all nonzero x2+y2).

See Heaviside step function § Analytic approximations.

The signum function x is non-zero: x<0, where it equals +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 x=0 in the ordinary sense, under the generalized notion of differentiation in distribution theory, the derivative of the signum function is two times the Dirac delta function. This can be demonstrated using the identity H(0)=12 formalism. Using this identity, it is easy to derive the distributional derivative: b, even when the interval of integration includes zero. The resulting integral for ∫ab(sgn⁡x)dx=|b|−|a|.

In fact, the signum function is the derivative of the absolute value function, except where there is an abrupt change in gradient at zero: x>0 and x>0, whose derivative is the constant value sgn⁡(0)=0. A subderivative value ∫sgn⁡(x)f(x)dx=sgn⁡(x)F(x)+Bsgn⁡(x)+C,where B can take on any value, most commonly 0. When considering the Dirac delta function, ∫sgn⁡(x)f(x)dx=sgn⁡(x)F(x)−∫2δ(x)F(x)dx,where H(x) is the Heaviside step function. Since F(0) can take on any value as long as it is defined. If Dirac delta function is considered, the integration constant of PV means taking the Cauchy principal value.

The signum function can be generalized to complex numbers as: z=0. The signum of a given complex number z≠0, z=0: csgn⁡z={1if Re(z)>0,−1if Re(z)<0,sgn⁡Im(z)if Re(z)=0 where Im(z) is the imaginary part of csgn⁡z=zz2=z2z.

Thanks to the Polar decomposition theorem, a matrix 𝕂∈{ℝ,ℂ}) can be decomposed as a product 𝑷 is a self-adjoint, or Hermitian, positive definite matrix, both in 𝑸 plays the role of 𝑹 is unitary, but generally different than 𝑹.

In the special case where a+ib=c, then the signum matrices satisfy sgn⁡c=c/|c|. In this sense, polar decomposition generalizes to matrices the signum-modulus decomposition of complex numbers.

At real values of ε(x)2=1 everywhere, including at the point (sgn⁡0)2=0. 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 x=0; and the special name, ε(0) is not defined, but Weisstein, Eric W. "Heaviside Step Function". MathWorld.