Generalized Eigenvectors

Generalized Eigenvector Definition

A vector \(v_m\)is a generalized eigenvector of rank m of a matrix \(A\) and corresponding to the eigenvalue \(\lambda\) if

$$(A-\lambda I)^mv_m=0$$

But

$$(A-\lambda I)^{m-1}v_m\neq 0$$

예시

Chain of generalized eigenvectors of length m=3

m = 3이고

$$(A-\lambda_1 I)^3v=0$$

일 때

$$\begin{align}v_3&:=v\\v_2&:=(A-\lambda_1I)v_3\\v_1&:=(A-\lambda_1I)v_2=(A-\lambda_1I)^2v_3\end{align}$$

으로 놓으면

$$\begin{align}(A-\lambda_1 I)^3v_3&=0\\(A-\lambda_1 I)^2v_2&=(A-\lambda_1I)^3v_3=0\\(A-\lambda_1I)v_1&=(A-\lambda_1 I)^3v_3=0\end{align}$$

으로 generalized eigenvector의 정의에 맞는 것을 볼 수 있다.
Jordan form으로의 적용을 위해 전개하면

$$\begin{align}Av_3&=v_2+\lambda_1v_3\\Av_2&=v_1+\lambda_1 v_2\\Av_1&=\lambda_1v_1\end{align}$$

이 된다.