For linear continuous dynamical systems, is the only possible equilibrium point 0?

Question

For linear continuous dynamical systems, is the only possible equilibrium point 0? In all the examples I have seen, only the 0 equilibrium point is considered. I know this is not true for nonlinear continuous dynamical systems.

Answer 1

Yes, it is for every hyperbolic (nonzero eigenvalues) linear system $\dot{x} = Ax$. Just solve the equation $A x = 0$.

Answer 2

It is not. All the points in the null space of $A$ are equilibrium points. The null space of $A$ is $\{0\}$ if and only if $A$ is invertible. For example take

$$A = \begin{bmatrix}-1 & 1 \\ 0 & 0\end{bmatrix}$$

So, the null space of $A$ is the space spanned by the vector $[0 ~~ 1]^T$, so all the points $[0 ~~ \alpha]^T$ for all $\alpha \in \mathbb{R}$ are equilibrium points.