It's already in its jordan form. So if you want .
Hello, everyone. I got a problem from my coursework question which relevent to controllability.
The question asks that use to show that a system dx/dt=AX+BU for which and is controllable whatever values c and d, take except d = 0.
I guess that the matrix A should be converted to a Jordan form, then ? But how to achieve and jordan matix in this question?
Thanks for help.
As I am not entirely sure of where your understanding of why you use the Jordan form, I thought I would add a little more for ya.
The important thing about the Jordan Form is that (provided your field contains all the eigenvalues, if not you can just pass to an algebraic closure and work there instead) you can write your Jordan Matrix J=D+N.
D is a diagonal matrix and N is a nilpotent matrix. These are particularly important because it greatly reduces your computations when doing things like oh, I don't know, taking powers of a matrix! Which makes this method incredibly desirable for writing out the exponential of a matrix.
So , but it gets even better because for powers of A you see all the interior P's cancel out and you get:
This is really easy to calculate since D and N commute you can use binomial theorem like you would hope (this takes some work to show). D is diagonal so any power of D is just the powers of the diagonal and for N, it is nilpotent, so at some point it just terminates completely and is just 0 from then on.
In your case
And
I was thinking that my question may be similar to this example, shows below:
Then I thought A should be converted to a Jordan form first, just like the above example did. But as you said if the matrix A already being a Jordan form, then how can I show that the system is controllable?Consider the system , which has matrix form:
for which we can show that , where
, , and .
Then
Hence .
Suppose now that there is a vector
such that .
From the constant part we see that , while from the coefficient of it follows that .
Since cannot be non-zero, we have shown that the above system is controllable.