Eigenstructure

Eigenstructure refers to the collection of eigenvalues and eigenvectors that characterize a square matrix or linear operator. More precisely, it comprises the spectrum of the operator and the corresponding eigenvectors, and often includes left eigenvectors and, in cases of non-diagonalizable matrices, generalized eigenvectors and the Jordan form. The right eigenvectors satisfy A v = λ v, while left eigenvectors satisfy w^T A = λ w^T. When a matrix has enough linearly independent eigenvectors to form a basis, it is diagonalizable and can be written as A = V Λ V^{-1}, where V contains the eigenvectors and Λ is a diagonal matrix of eigenvalues.

The eigenstructure determines how a linear transformation acts on vectors and provides a natural modal decomposition of the system. Each eigenpair (λ, v) represents a mode that grows, decays, or oscillates independently over time, scaled by λ. This information is central to stability analysis, dynamic response, and control design. In nonlinear systems, linearization around an equilibrium yields a Jacobian, and the eigenstructure of that Jacobian predicts local behavior near the fixed point.

In practice, the eigenstructure informs several techniques: modal analysis in engineering, spectral methods in numerical linear algebra, and observer or controller design in control theory. Challenges include handling complex eigenvalues, non-diagonalizable (defective) matrices requiring generalized eigenvectors, and numerical sensitivity in computing the full eigenstructure. Specialized algorithms (such as QR iteration and other spectral methods) strive to accurately and efficiently extract eigenvalues and eigenvectors for large-scale problems.