Eigenvalues and eigenvectors

Eigenvalues and eigenvectors are mathematical concepts used to analyze linear transformations. An eigenvector is a special vector that, when a transformation (like stretching or rotating) is applied to it, only changes in size (scales) but not direction. The eigenvalue corresponds to the factor by which the eigenvector is stretched or compressed during this transformation. In essence, eigenvectors reveal directions that remain consistent, while eigenvalues quantify how much these directions are scaled, helping us understand the structure and behavior of complex systems like matrices, data, or physical processes.