Why for a linear system, the stability for a generic equlibrium point is equivalent to the stability of the origin?

Question

I am studying the concept of stability for linear and for nonlinear systems. While studying the stability for a linear system I found this definition from the notes of my professor:

for a linear system, the stability for a generic equlibrium point is equivalent to the stability of the origin.

can somebody explain to me what does it mean?

Answer 1

A linear system has the equation

$$ \dot{x} = A x $$

and we assume that this system is stable. The equilibrium are the solution of $A x = 0$. This is a linear equation system.

Think of the points:

1. We know that a linear equation system $A x = b$ has a unique solution if and only if $A$ is nonsingular (in our case $b = 0$ but that doesn't change anything).
2. We know that all eigenvalues of $A$ have negative real part if and only if the linear system is stable (which is what we assumed). This means the eigenvalues are nonzero.
3. We know that a matrix $A$ can be singular if and only if it has at least one eigenvalue equal to zero.
4. We know that $x = 0$ is always a solution of $A x = 0$.

We can put it together: Because $\dot{x} = A x$ is stable the eigenvalues of $A$ are nonzero. Because the eigenvalues are nonzero, $A$ is nonsingular. Because $A$ is nonsingular $A x = 0$ has a unique solution. Because $x = 0$ is always a solution of $A x = 0$ it is always the only solution.

Conclusion: If $\dot{x} = A x$ is stable then its only equilibrium can be the origin.