# mathlibdocumentation

linear_algebra.eigenspace

# Eigenvectors and eigenvalues #

This file defines eigenspaces, eigenvalues, and eigenvalues, as well as their generalized counterparts. We follow Axler's approach [Axl15] because it allows us to derive many properties without choosing a basis and without using matrices.

An eigenspace of a linear map f for a scalar μ is the kernel of the map (f - μ • id). The nonzero elements of an eigenspace are eigenvectors x. They have the property f x = μ • x. If there are eigenvectors for a scalar μ, the scalar μ is called an eigenvalue.

There is no consensus in the literature whether 0 is an eigenvector. Our definition of has_eigenvector permits only nonzero vectors. For an eigenvector x that may also be 0, we write x ∈ f.eigenspace μ.

A generalized eigenspace of a linear map f for a natural number k and a scalar μ is the kernel of the map (f - μ • id) ^ k. The nonzero elements of a generalized eigenspace are generalized eigenvectors x. If there are generalized eigenvectors for a natural number k and a scalar μ, the scalar μ is called a generalized eigenvalue.

## Tags #

eigenspace, eigenvector, eigenvalue, eigen