for the following matrices A, classify the stability of the linear systems x=Ax as asymptotically stable, L-stable (but not asymptotically stable) or unstable and indicate whether it is a stable node, stable degenerate node, etc:
I don't really know how to get started…can someone explain what the first step would be?
So someone said that I should look at the eigenvalues and:
a.
\begin{array}{cc}
-6 & -3 \\
2 & -1 \\
\end{array}
$λ_1=-3$, $λ_2=-4$
Asymptotically stable
Node
b.
\begin{array}{cc}
5 & -5 \\
2 & -1 \\
\end{array}
$λ_1=2+i$, $λ_2=2-i$
Unstable
Spiral Point
c.
\begin{array}{cc}
1 & 5 \\
-2 & -1 \\
\end{array}
$λ_1=3i$, $λ_2=-3i$
L-stable
Center
d.
\begin{array}{cc}
1 & -2 \\
8 & -7 \\
\end{array}
$λ_1=-3$, $λ_2=-3$
Asymptotically Stable
Proper Node
Best Answer
For continuous linear time-invariant systems like this, you can determine stability by looking at the eigenvalues of the matrix $A$.
If the real part of each eigenvalue is strictly negative, the system is asymptotically stable. If some eigenvalues have negative real part but one or more of them has zero real part, the system is marginally stable but not asymptotically stable. If any eigenvalue has positive real part, the system is unstable.
Can you take it from here?