Eigenvalues and Eigenvectors are fundamental concepts in linear algebra, playing a critical role in various mathematical, engineering, and scientific applications. For a given square matrix A, an eigenvector is a non-zero vector v that, when multiplied by $A$, results in a scalar multiple of itself. This relationship is expressed as Av=λv, where $\lambda$ is the eigenvalue corresponding to the eigenvector v. In simpler terms, applying the matrix  $A$ to the eigenvector v does not change its direction, only its magnitude, which is scaled by the eigenvalue λ.

Eigenvalues and eigenvectors are essential in understanding the properties of linear transformations, such as rotation, reflection, and scaling. They are widely used in numerous fields, including data science, physics, computer graphics, and machine learning, where they help simplify complex systems, reduce dimensionality, and identify patterns in data. The process of finding eigenvalues and eigenvectors involves solving the characteristic equation, det⁡(A−λI)=0 , where I is the identity matrix, and λ represents the possible eigenvalues.