Eigenvectors, eigenvalues, and eigenspaces

Generalized eigenvector

A generalized eigenvector fulfils a more relaxed condition than a regular eigenvector. If $𝑉$ is a $𝕂$ -vector space and $𝐴 \in E n d 𝕂 𝑉$ , then a vector $𝑤 \in 𝑉$ is a generalized eigenvector of rank $𝑚$ and eigenvalue $𝜆$ iff

(𝐴 - 𝜆 1 𝑉) 𝑚 𝑤 = ⃗ 𝟎

and $𝑚 \in ℕ$ is the minimum integer such that the equation is satisfied. Thus regular eigenvectors are subsumed as generalized eigenvectors of rank 1. When the rank is left unspecified, the requirement is rather that there exists some $𝑚 \in ℕ$ for which the above holds, and we define

˜ E 𝜆 𝑉 = {𝑣 \in 𝑉 : (\exists 𝑚 \in ℕ) [(𝐴 - 𝜆 1 𝑉) 𝑚 = 0]}

as the generalized eigenspace

Finding eigenvectors of rank 2

For generalised eigenvectors of rank 2, if a regular eigenvector $⃗ 𝐯 \in 𝐸 𝜆 (𝐴)$ is already known, the following equation can be solved for $⃗ 𝐰$

(𝐴 - 𝜆 𝐼) ⃗ 𝐰 = ⃗ 𝐯

Properties

Generalized eigenspace decomposition

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