The spectral radius of a matrix is defined as the largest absolute value of its eigenvalues. Eigenvalues are special numbers associated with a matrix that provide insights into its properties, such as stability and behavior under transformations. The spectral radius helps in understanding how a matrix will act when raised to high powers or when applied to vectors. In practical applications, the spectral radius is crucial in fields like control theory, numerical analysis, and graph theory. It can indicate whether a system is stable or unstable, guiding engineers and scientists in designing systems that behave predictably.