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

$$(A-\lambda I{)}^m{v}_m=0$$

But

$$(A-\lambda I{)}^{m-1}{v}_m\ne 0$$

예시

Chain of generalized eigenvectors of length m=3

m = 3이고

$$(A-{\lambda }_{1}I{)}^{3}v=0$$

일 때

$$\begin{array}{rl}{v}_3& :=v\\ v_2& :=(A-{\lambda }_{1}I)v_3\\ v_1& :=(A-{\lambda }_{1}I)v_2=(A-{\lambda }_{1}I{)}^2{v}_3\end{array}$$

으로 놓으면

$$\begin{array}{rl}(A-{\lambda }_{1}I{)}^3{v}_3& =0\\ (A-{\lambda }_{1}I{)}^2{v}_2& =(A-{\lambda }_{1}I{)}^3{v}_3=0\\ (A-{\lambda }_{1}I)v_1& =(A-{\lambda }_{1}I{)}^3{v}_3=0\end{array}$$

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

$$\begin{array}{rl}A{v}_3& =v_2+{\lambda }_1{v}_3\\ A{v}_2& =v_1+{\lambda }_1{v}_2\\ A{v}_1& ={\lambda }_1{v}_1\end{array}$$

이 된다.