Quasieigenpairs

Quasieigenpairs are a concept in linear algebra and numerical analysis that generalize the notion of eigenpairs (eigenvalues and eigenvectors) to matrices that are not necessarily diagonalizable. An eigenpair (λ, v) of a matrix A satisfies the equation Av = λv, where λ is the eigenvalue and v is the corresponding eigenvector. However, for matrices that do not have a complete set of linearly independent eigenvectors, the concept of quasieigenpairs is used.

A quasieigenpair (λ, v) of a matrix A satisfies the equation (A - λI)v = εr, where ε is a small perturbation and r is a residual vector. The vector v is called a quasieigenvector, and λ is called a quasieigenvalue. Quasieigenpairs are useful in numerical analysis because they provide a way to approximate the eigenvalues and eigenvectors of a matrix, even when the matrix is not diagonalizable.

Quasieigenpairs can be computed using various numerical methods, such as the QR algorithm with shifts or the Arnoldi iteration. These methods iteratively refine the quasieigenpairs until they converge to the true eigenpairs or until a desired level of accuracy is achieved.

Quasieigenpairs have applications in various fields, including physics, engineering, and computer science. For example, in quantum mechanics, quasieigenpairs can be used to approximate the energy levels of a quantum system. In control theory, quasieigenpairs can be used to analyze the stability of a linear system. In computer science, quasieigenpairs can be used to solve large sparse linear systems.