Eigenvalues are really important when we look at how stable dynamic systems are. This is especially true for linear differential equations.

So, what does stability mean?

It means that if we make a small change at the start, we want to know how it will affect the system over time.

When we study a linear system, we often use a differential equation that looks like this:

$\frac{d\mathbf{x}}{dt} = A\mathbf{x},$

Here, $A$ is a matrix. The eigenvalues of matrix $A$ give us important information about stability.

We can check how stable the point where $\mathbf{x} = \mathbf{0}$ (which we call the equilibrium point) is by looking at the real parts of the eigenvalues. Here’s how it works:

• If all eigenvalues ($\lambda_i$) have negative real parts ($\text{Re}(\lambda_i) < 0$), the system is stable. This means that over time, it will move closer to the equilibrium point.

• If any eigenvalue has a positive real part ($\text{Re}(\lambda_i) > 0$), the system is unstable. In this case, it will move away from the equilibrium point.

• If the eigenvalues have zero real parts, we need to look more closely, as the system might behave neutrally or go back and forth without settling.

In simple terms, eigenvalues help us predict how a dynamic system will behave in the long run. By understanding them, we can tell if a system is stable, unstable, or critically stable. This knowledge is useful for creating and controlling many applications in engineering, physics, and more.