quasieigenpair

In linear algebra, a quasieigenpair is a concept that arises in the context of matrices and their eigenvalues and eigenvectors. It is closely related to the concept of eigenpairs, but with a slightly different interpretation. A quasieigenpair is a pair of elements, typically a scalar and a non-zero vector, that satisfy a matrix's eigenvector and eigenvalue equations, albeit with a relaxed condition.

More formally, given a square matrix A of size n x n and a pair of scalars λ and μ, along with an n x 1 vector v, the quasieigenpair condition is met if the following equation holds:

(λ - μ)Av = v

Thus, a quasieigenpair consists of a scalar λ and a vector v that satisfy the above equation for some scalar μ, possibly different from λ. This is in contrast to an eigenpair, where the eigenvalue and eigenvector are directly related.

Quasieigenpairs have practical importance in various mathematical and computational applications, such as perturbation theory, where small changes to a matrix are studied. The concept of quasieigenpairs generalizes to matrices of any size and provides a more nuanced understanding of a matrix's properties, particularly when dealing with very large or complex matrices.

The study of quasieigenpairs has led to a deeper understanding of matrices and their behavior under different conditions. Researchers have shown that certain results derived for eigenpairs can be applied, albeit with additional assumptions, to the broader class of quasieigenpairs. This enables further investigation into various areas of linear algebra and has potential applications in optimization, numerical analysis, and other fields.