I need to check that I'm solving this question correctly and where to go afterwards. I'm reducing the given equation to a controllable and observable equation.
From the picture, it is clear the A matrix is in Jordan form where λ1 is in a 3×3 Jordan block and λ2 is in a 2×2 Jordan block. The B matrix that corresponds to λ1 is the top 3 elements. Since the third element is 0, λ1 is not controllable.
The B matrix that corresponds to λ2 is the bottom 2 elements. Since the 5th element is 1, it is controllable.
For Observability, I need to check the C matrix. Since the first element is 0, λ1 is not observable, and since the fourth element is 0, λ2 is not observable.
So I know λ1 is not controllable or observable and λ2 is controllable but not observable. How do I write the corresponding state equation?