I have started to learn dynamic systems and I am stuck on the following problem.
X(t) = [1, 12] * [x1]
---------3, 1------ x2
I find the eigen values to be 7 and -5, which is correct and now try to find the eigenvectors.
I am using the Jordan decomposition method such that the solution eAt=PeRtP-1
where R is a diagonal matrix whose diagonal elements are eigenvalues of A and P is a matrix whos columns are the eigenvectors of A.
I am struggling now to find matrix P.
I know that I need to solve (A-7I)(V1,V2) = 0 and (A+5I)(V1,V2) = 0 where v1 and v2 are column vectors.
However when I try to do this I get a system of equations with two unknowns, for example for (A-7I)(V1,V2) = 0
We are solving [-6, 12](V1,V2)=0 I am unsure how to find a solution for this case
Apparently the answer is V=(2,1) for the first case and V=(-2,1) for the second case, if someone could establish this link for me I would greatly appreciate it.
Thank you very much!