Context: probability theory, statistics

In probability theory, the characteristic function (φ(s)) of a random variable X is defined as the expected value of eisX, where e is the base of the natural logarithm, and i is the imaginary number.

If X comes from an absolutely continuous distribution, i.e. one that can be completely specified by a probability density function, then the characteristic function of X is the same as the Fourier transform of the density of X.

Unlike the moment generating function, which is not defined for subexponential distributions, the characteristic function exists for all distributions, and shares many of the same properties of moment generating functions.

The characteristic equation for an nxn matrix A is the equation det(A – λ ⋅ I)=0. The left hand side of this equation (det(A – λ ⋅ I)) is called the characteristic polynomial. There will be n values of λi (called eigenvalues) that satisfy this equation that may not be distinct, each corresponding to an eigenvector (which might not be unique) vi that satisfies the equation Avi = λivi. According to the Cayley-Hamilton Theorem, a matrix A satisfies its own characteristic equation.

The characteristic equation for a linear difference or differential equation is generated by converting the equation into a system of linear first order equations, and finding the characteristic equation of the coefficient matrix. For example:

y'' – 2y' + y = 0    Let y = x1, y' = x2

x1' = x2
x2' = 2y' – y = 2x2 - x1

x' = Ax, where x = (x1, x2) and A = (0  1)
(-1 2)

det(A – λ ⋅ I) = –λ(2 – λ) + 1 = 0

The characteristic equation of the differential equation is λ2 – 2λ + 1 = 0. Another way to look at it is to replace the y's with λ's and convert the derivatives into exponents, or for difference equations convert the indices to exponents and divide out any excess multiples of λ.

Sources:
Eric W. Weisstein. "Characteristic Equation." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/CharacteristicEquation.html

jrn says re characteristic equation : A small point perhaps worth mentioning is that if you substitute y=e^(ax) into a differential equation you get the characteristic polynomial times e^(ax).

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