Eigenvalues & Eigenvectors

In this chapter, we introduce the concepts of eigenvalues and eigenvectors of a square matrix. These concepts have numerous uses, for example, to better understand and visualize linear mappings, to understand the stability of mechanical constructions, for solving systems of differential equations, to recognize images, to interpret and visualize quadratic equations, and for image segmentation. Let A be a square matrix. A non-zero column vector v is called an eigenvector if Av is parallell to v, i.e., if there exists a scalar λ such that
The scalar λ is then called an eigenvalue.