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)
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.
Eric W. Weisstein. "Characteristic Equation." From MathWorld--A Wolfram Web Resource.
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).