Eigenvalue

Eigenvalues are special numbers associated with a matrix that reveal how the matrix transforms space. When a matrix acts on a vector aligned in a certain direction, that vector is scaled by its eigenvalue but not rotated. Think of it like a filter: some directions are stretched or shrunk by specific factors (the eigenvalues), just as a lens magnifies or reduces certain features. Eigenvalues help us understand the intrinsic properties of a system, such as stability and natural modes, and are fundamental in fields like engineering, physics, and data analysis.