Sign function

In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zero. In mathematical notation the sign function is often represented as \sgn x or \sgn (x) .

In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zero. In mathematical notation the sign function is often represented as \sgn x or \sgn (x) . ==Definition== The signum function of a real number x is a piecewise function which is defined as follows: \sgn x :=\begin{cases} -1 & \text{if } x 0. \end{cases} 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 , or which can then be used in mathematical expressions or further calculations. For example: \begin{array}{lcr} \sgn(2) &=& +1\,, \\ \sgn(\pi) &=& +1\,, \\ \sgn(-8) &=& -1\,, \\ \sgn(-\frac{1}{2}) &=& -1\,, \\ \sgn(0) &=& 0\,. \end{array} x = |x| \sgn x\,. It follows that whenever x is not equal to 0 we have \sgn x = \frac{x} = \frac {x}\,. Similarly, for any real number x , |x| = x\sgn x\,. We can also be certain that: \sgn (xy)=(\sgn x)(\sgn y)\,, and so \sgn (x^n)=(\sgn x)^n\,. From the following: \frac{x} \times \frac{x} = \frac{x} \times \frac {x}\ = 1\qquad \text{for } x \ne 0\,, the following two statements hold true for any integer n : \left ( \sgn x \right )^{2n} = 1\qquad \text{for } x \ne 0\,, \left ( \sgn x \right )^{2n-1} = \sgn x\qquad \text{for } x \ne 0\,. Hence also: \left ( \sgn x \right )^{\pm\sgn x} = \sgn x\qquad \text{for } x \ne 0\,. \sgn x = [x > 0] - [x The signum can also be written using the floor and the absolute value functions: \sgn x = \Biggl\lfloor \frac{x} {\text{d}x} = \sgn x \qquad \text{for } x \ne 0\,. We can understand this as before by considering the definition of the absolute value |x| on the separate regions x>0 and x For example, the absolute value function is identical to x in the region x>0, whose derivative is the constant value , which equals the value of \sgn x there. Because the absolute value is a convex function, there is at least one subderivative at every point, including at the origin. Everywhere except zero, the resulting subdifferential consists of a single value, equal to the value of the sign function. In contrast, there are many subderivatives at zero, with just one of them taking the value \sgn(0) = 0 . A subderivative value occurs here because the absolute value function is at a minimum. The full family of valid subderivatives at zero constitutes the subdifferential interval [-1,1] , which might be thought of informally as "filling in" the graph of the sign function with a vertical line through the origin, making it continuous as a two dimensional curve. In integration theory, the signum function is a weak derivative of the absolute value function. Weak derivatives are equivalent if they are equal almost everywhere, making them impervious to isolated anomalies at a single point. This includes the change in gradient of the absolute value function at zero, which prohibits there being a classical derivative. The antiderivative of a function multiplied by the signum function is: \int {\sgn (x) f(x) \text{d}x} = \sgn (x) F(x) + B\sgn (x) + C \,, where B and C are constants of integration. Outside of distribution theory, B can take on any value, most commonly 0. When considering the Dirac delta function, B has to equal F(0) . This can be proven using integration by parts: \int {\sgn (x) f(x) \text{d}x} = \sgn(x) F(x) - \int {2 \delta(x) F(x)\text{d}x}\,, where \delta(x) is the Dirac delta function. Integrating, the following expression can be obtained: \sgn(x) F(x) - \int {2 \delta(x) F(x)\text{d}x} = \sgn (x) F(x) - 2F(0)H(x) + C = \sgn (x) F(x) - \sgn(x) F(0) + C \,, where H(x) is the Heaviside step function. Since F(x) already has its own integration constant, unrelated to C , F(0) can take on any value as long as it is defined. If Dirac delta function is considered, the integration constant of F(x) has to mach in both terms; otherwise, the derivative of the signum function is 0, so it can be multiplied by any constant in the second term. PV\int_{-\infty}^\infty (\sgn x) e^{-ikx}\text{d}x = \frac{2}{ik} \qquad \text{for } k \ne 0, where PV means taking the Cauchy principal value. \sgn z = \frac{z} for any complex number z except z=0 . The signum of a given complex number z is the point on the unit circle of the complex plane that is nearest to z . Then, for z\ne 0 , \sgn z = e^{i\arg z}\,, where \arg 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 z=0 : \sgn(0+0i)=0 Another generalization of the sign function for real and complex expressions is \text{csgn} , which is defined as: \operatorname{csgn} z= \begin{cases} 1 & \text{if } \mathrm{Re}(z) > 0, \\ -1 & \text{if } \mathrm{Re}(z) where \text{Re}(z) is the real part of z and \text{Im}(z) is the imaginary part of z . We then have (for \mathrm{Re}(z)\ne 0 ): \operatorname{csgn} z = \frac{z}{\sqrt{z^2}} = \frac{\sqrt{z^2}}{z}. Thanks to the Polar decomposition theorem, a matrix \boldsymbol A\in\mathbb K^{n\times n} ( n\in\mathbb N and \mathbb K\in\{\mathbb R,\mathbb C\} ) can be decomposed as a product \boldsymbol Q\boldsymbol P where \boldsymbol Q is a unitary matrix and \boldsymbol P is a self-adjoint, or Hermitian, positive definite matrix, both in \mathbb K^{n\times n} . If \boldsymbol A is invertible then such a decomposition is unique and \boldsymbol Q plays the role of \boldsymbol A 's signum. A dual construction is given by the decomposition \boldsymbol A=\boldsymbol S\boldsymbol R where \boldsymbol R is unitary, but generally different than \boldsymbol Q . This leads to each invertible matrix having a unique left-signum \boldsymbol Q and right-signum \boldsymbol R . In the special case where \mathbb K=\mathbb R,\ n=2, and the (invertible) matrix \boldsymbol A = \left[\begin{array}{rr}a&-b\\b&a\end{array}\right] , which identifies with the (nonzero) complex number a+\mathrm i b=c , then the signum matrices satisfy \boldsymbol Q=\boldsymbol P=\left[\begin{array}{rr}a&-b\\b&a\end{array}\right]/|c| and identify with the complex signum of c , \sgn c = c/|c| . In this sense, polar decomposition generalizes to matrices the signum-modulus decomposition of complex numbers. At real values of x , it is possible to define a generalized function–version of the signum function, \varepsilon (x) such that \varepsilon (x)^2=1 everywhere, including at the point x=0 , unlike \sgn , for which (\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 \varepsilon (x) \delta(x)+\delta(x) \varepsilon(x) = 0 \, ; in addition, \varepsilon (x) cannot be evaluated at x=0 ; and the special name, \varepsilon is necessary to distinguish it from the function \sgn . ( \varepsilon (0) is not defined, but \sgn 0=0 .) ==See also== Absolute value Heaviside step function Negative number Rectangular function Sigmoid function (Hard sigmoid) Step function (Piecewise constant function) Three-way comparison Zero crossing Polar decomposition Parity of a permutation ==Notes==

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