Eigenvalues and differential equations

In my view, the most interesting application of eigenvalues and vectors is Principal Component Analysis, a method for determining the most significant dimensions in a data set. However, since I already wrote about that, I'll take a quick look at some more traditional applications.

One of the classic uses is in solving differential equations when the value at some point is known (generally referred to as initial value problems). Here, the formulation is:



where y_i = f_i(t) is a continuous function over the relevant domain of t.

In vector form, this becomes Y' = AY and the solution will be of the form Y = e^λtx. Thus, if λ is an eigenvalue of A then AY = e^λtAx = λe^λtx = λY = Y'. So, the eigenvectors of A provide a basis for the solution space of continuous vector-valued functions satisfying the conditions. To force a unique solution, an additional constraint must be added. Setting Y(0) = Y₀ allows one to solve exactly. This is considered an the initial value, though it doesn't technically have to occur at time 0 as once can transform the input to use a value at any known time.

This holds whether the eigenvalues are real or complex and can be generalized to higher-order systems by partitioning the matrix A.