eigen

Eigen is a term in linear algebra and a name used by a widely used C++ template library for linear algebra. In mathematics, an eigenvalue and its corresponding eigenvector describe a special relation for a square matrix A: Av = λv, where λ is a scalar and v is a nonzero vector. The scalar λ is called an eigenvalue, and the vector v an eigenvector of A. The eigenvalues satisfy the characteristic equation det(A − λI) = 0. If A has a complete set of eigenvectors, it can be diagonalized as A = VΛV⁻¹, with Λ containing the eigenvalues and the columns of V the eigenvectors. For real symmetric or complex Hermitian matrices, eigenvalues are real and eigenvectors can be chosen orthonormal, reflecting the spectral theorem.

Eigenvalues and eigenvectors have broad applications, including principal component analysis, stability analysis in differential equations, vibration modes in mechanical systems, and the long-run behavior of Markov chains. They also underpin algorithms in data analysis and computer graphics. Computing eigenpairs typically involves iterative methods such as the QR algorithm or power iteration, with specialized solvers for large sparse or symmetric matrices.

Eigen is also the name of a popular open-source C++ template library for linear algebra. It is primarily header-only and supports dense and sparse matrices and vectors, along with a wide range of decompositions and solvers. Features include LLT/LDLT, LU, QR, JacobiSVD, BDCSVD, EigenSolver, SelfAdjointEigenSolver, and GeneralizedEigenSolver, among others. The library emphasizes expressive syntax, performance through expression templates and vectorization, and broad platform support, making it a common choice in scientific computing and graphics.