Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are central to linear algebra, serving as powerful tools to decode the intrinsic behavior of linear transformations. The term ’eigen’ (German for ’own’) was introduced by David Hilbert in 1904 in the context of integral equations, though the underlying concepts trace back to earlier work by Euler, Cauchy, and others. This property is akin to finding the “natural axes” of a transformation, simplifying complex matrix operations into scaling along specific directions. In ML, eigenvalues and eigenvectors underpin algorithms like PCA, spectral clustering, and optimization techniques, making them indispensable for data analysis and model design.