Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra, widely used in various fields, including machine learning, data science, and physics. An eigenvalue is a scalar that indicates how much a corresponding eigenvector is stretched or compressed during a linear transformation represented by a matrix. In other words, if a matrix acts on an eigenvector, the output is the eigenvector scaled by the eigenvalue. These concepts are crucial for dimensionality reduction techniques like Principal Component Analysis (PCA), where they help identify the directions of maximum variance in high-dimensional data. Additionally, they play a significant role in stability analysis and systems dynamics.