eigencomponent

Eigencomponent is a term used in linear algebra to denote the component of a vector along a given eigenvector of a linear transformation or matrix. When a matrix A has a complete set of eigenvectors (i.e., is diagonalizable), any vector x can be written as a linear combination of the eigenvectors: x = sum_i c_i v_i, where v_i are the eigenvectors of A and c_i are scalar coefficients. In this sense, the eigencomponent associated with eigenvalue λ_i is the vector c_i v_i, the portion of x that lies in the eigen-direction corresponding to λ_i.

Computation follows from the diagonalization of A. If A = P D P^{-1} with D diagonal and the columns of P being the eigenvectors v_i, then x = P c for c = P^{-1} x, and the i-th eigencomponent is the scalar c_i times v_i. In the special case that A is symmetric, its eigenvectors can be chosen orthogonal, giving c_i = (v_i^T x) / (v_i^T v_i). If A is not diagonalizable, an eigenbasis may not exist and the simple projection interpretation breaks down; one can instead use generalized eigenvectors and Jordan form, where components are defined with respect to a Jordan basis rather than a complete eigenbasis.

Applications of eigencomponents appear in modal analysis, spectral decomposition, and state-space analysis, where they describe how much of a state or signal lies in each eigenmode. For a matrix with a full eigenbasis, the space decomposes into invariant subspaces and x can be reconstructed as the sum of its eigencomponents. Example: A = diag(2, 3) with eigenvectors e1, e2; for x = [x1, x2], the eigencomponents are x1 e1 and x2 e2.