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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.
Eigenvector v
Knowing , we will proceed to find our eigenvectors by solving .
which gives the homogeneous system
.
Placing this in reduced-row form we get
![{\displaystyle \left[{\begin{array}{cc}2&1\\0&0\end{array}}\right]}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/f49814a25c19b97c1f65a13d82a6d42da99c04d7)
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

where
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

where

We can therefore write,

where
.
Solve the simplified system y' = My
The system

tells us
.
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,
.
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