In the context of dynamical systems or linear stability analysis, the terms "stable" and "unstable" are used to describe the behavior of eigenvalues. Let's first define what an eigenvalue is in this context.
In linear algebra, given a matrix, an eigenvalue represents a scalar value associated with that matrix. For a linear transformation, an eigenvalue corresponds to a value for which a non-zero vector, called an eigenvector, remains in the same direction (up to a scalar multiple) when transformed by the matrix.
Now, when analyzing the stability of a dynamical system, such as a differential equation or a difference equation, the eigenvalues of the system's matrix play a crucial role. The stability of a system is determined by the behavior of its eigenvalues.
Stable eigenvalues: An eigenvalue is considered stable if, when associated with a dynamic system, it leads to a behavior where small perturbations or disturbances dampen out over time. In other words, if the system starts near an equilibrium point corresponding to a stable eigenvalue, it will tend to return to that point and remain there. Stability implies that the system's behavior is predictable and converges towards a steady state.
Unstable eigenvalues: An eigenvalue is classified as unstable if small perturbations grow over time instead of diminishing. In this case, the system's behavior becomes divergent or unpredictable. Even slight deviations from an equilibrium point corresponding to an unstable eigenvalue can cause the system to move away from that point exponentially.
It's important to note that stability or instability can have different meanings depending on the specific context. For example, in the context of stability analysis of numerical methods for solving differential equations, stable eigenvalues refer to those that do not cause numerical solutions to grow and become inaccurate over time.
In summary, stable eigenvalues lead to predictable and convergent behavior, while unstable eigenvalues result in divergent or unpredictable behavior in the context of dynamical systems.