I am trying to understand linear stability:
- A linear system is linearly stable if all its solutions are bounded as $t \to \infty$.
I have the following linear systems
$x'(t)=Ax(t)$ and $y'(t)=By(t)$ where
$A= \left(\begin{array}{cc} 0 & 1\\ -1 & 0 \end{array}\right)$ and $B=\left(\begin{array}{cc} 2 & 1\\ 0 & -4 \end{array}\right)$
I know they are $A$ is linearly stable and $B$ is not and I am trying to understand why.
In the explanation for $A$ it says since the eigenvectors are independent and from that we can prove that the nilpotent part of the matrix vanishes?
For $B$ I believe it is not linearly stable simply because one of the eigenvalues is positive.
Can anyone explain simply why $A$ is linearly stable to me? If there are no repeated eigenvectors does this normally imply the system is linealrly stable?
Thank you!