eigendecompositions

Eigendecomposition is a factorization of a square matrix into a set of eigenvectors and eigenvalues. For a given square matrix A, an eigendecomposition is represented as A = PDP⁻¹, where P is a matrix whose columns are the eigenvectors of A, D is a diagonal matrix containing the corresponding eigenvalues, and P⁻¹ is the inverse of P. This decomposition is only possible if the matrix A has a full set of linearly independent eigenvectors.

The eigenvectors are non-zero vectors that, when multiplied by the matrix A, are simply scaled by a factor. This scaling factor is the eigenvalue. Mathematically, for a non-zero vector v and a matrix A, if Av = λv, then v is an eigenvector and λ is its corresponding eigenvalue. Eigenvalues and eigenvectors are fundamental concepts in linear algebra and have wide-ranging applications.

Eigendecompositions are crucial in various fields. In physics, they are used to analyze the vibrations of systems and in quantum mechanics to determine energy levels. In statistics and machine learning, eigendecomposition is the basis for Principal Component Analysis (PCA), a dimensionality reduction technique. It is also used in graph theory to study the properties of networks and in solving systems of differential equations. The ability to transform a matrix into a simpler diagonal form via eigendecomposition reveals intrinsic properties of the linear transformation represented by the matrix.