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Linear Differential Equations

marcus2001

So basically I'm not sure of the relationship between the eigenvalues of A, where x'(t) = Ax(t) is a linear system of differential equations and it's solutions. (x'(t) and x(t) are n x 1 vectors and A an n x n matrix.)

For example, suppose the system is in R2 so A is a 2 x 2 matrix etc. Top row 3 2 bottom row 4 1. We've deduced the origin is an unstable point as one of the eigenvalues of A has real part greater than 0. We're trying to determine the line along which, if you started, you'd move to the origin (everywhere else you'd move away).

Apparently the direction you should begin is in the direction of the eigenvector corresponding to the eigenvalue with negative real part. The reasoning which I absolutely do not understand is that the eigenvector corresponding to the eigenvalue (of A) with negative real part is also an eigenvalue of e^tA (for all t) with corresponding eigenvalue e^-t.

Eternal reps to whoever can help me out!

(edited 9 years ago)

Reply 1

BlueSam3