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To begin with we matrix M in Jordan normal form, such that A = SMS-1. To do we need to find the (generalized) eigenvector of A.
Knowing , we will proceed to find our eigenvectors by solving .
which gives the homogeneous system
Placing this in reduced-row form we get
so that . We take
as the eigenvector.
Unfortunately, we only find one eigenvector from this, and we'll need to consider generalized eigenvectors and the Jordan canonical form in order to solve this problem.
Find the generalized eigenvector w
The generalized eigenvector satisfies which gives us
Row reductions lead to
which has a solution set that can be parameterized by :
For simplicity, we can take so that our generalized eigenvector is
Setting up the Jordan normal form M
Having found the generalized eigenvector, we get that the matrix A has Jordan Canonical form
- is the matrix with columns being the eigenvectors, and
- is the matrix with the eigenvalues along the diagonal with 1's above them.
A quick computation yields
Our system states that
We can therefore write,
Solve the simplified system y' = My
The second equation gives us
Using this in the first equation means
which has been solved using integrating factors. Now we have
Recover the solution x
which immediately gives us our solution x,