On the slide 7 of lecture 4.1, we claim that if all eigenvalues have non-positive real parts, then linear systems are stable. Actually, this is not correct.
For diagonal systems, this is correct because if a eigenvalue k with non-positive real part will generate exp(k*t) which is bounded.
For Jordan form, this is not correct. Suppose we have a Jordan block like this
0
1 0
Ji= 0 0 1
0 0 0
This means this system has multiple eigenvalues at 0. Suppose other eigenvalues are all real and negative. Then
exp(0*t) t*exp(0*t) 1/2*t2*exp(0*t)
1 t 1/2*t2
exp(Ji)=
0
exp(0*t)
t*exp(0*t) = 0
1 t
0
0
exp(0*t)
0 0 1
This Jordan block will result a polynomial w.r.t.time which goes to infinity as time increases. This system is unstable.
So we should be careful when we judge the stability of a linear system if it has multiple eigenvalues with zero real parts.