A characteristic equation is a mathematical expression derived from a linear differential equation or a matrix. It helps determine the behavior of a system, such as stability and oscillation, by finding the roots of the equation. These roots, known as eigenvalues, provide crucial information about the system's dynamics. In the context of control theory or linear algebra, characteristic equations are essential for analyzing systems. For example, in engineering, they are used to assess the stability of structures or electrical circuits. By solving the characteristic equation, engineers can predict how a system will respond to various inputs or disturbances.