applet-magic.com Thayer Watkins
Silicon Valley & Tornado Alley
U.S.A.

The Generalized Eigenvectors of a Matrix and their Linear Indepedence

Let A be an n×n matrix and let λ be an eigenvalue of A. Let 0 denote the vector of zeroes.
A nonzero vector X is said to be a generalized eigenvector of
rank m for A and eigenvalue λ if

(A-λI)^{m} = 0 but
(A-λI)^{m-1} ≠ 0

An ordinary eigenvector of A is also a generalized eigenvector of A because

(A-λI)X = 0 but
(A-λI)^{0}X = IX = X ≠ 0

The Chain of Vectors Generated from a Generalized Eigenvector of Rank m

Let X_{m} be a generalized eigenvector of rank m for matrix A corresponding to the eigenvalue λ.
Let X_{m-1} be created from X_{m} as (A-λI)X_{m} and likewise X_{m-2}
from X_{m-1} and so on down to X_{1}. Thus for j=1, … (m-1)

X_{j} = (A-λI)X_{j+1} and hence
X_{j} = (A-λI)^{m-j}X_{m}

The sequence {X_{m}, X_{m-1}, … X_{1}} is called the chain generated by
the generalized eigenvector X_{m}. Each element X_{j} of the chain is a generalized eigenvector
of A associated with its eigenvalue λ. Furthermore the rank of X_{j} is j.

Proof:

By definition

(A-λI)^{m}X_{m} = 0 but
(A-λI)^{m-1}X_{m} ≠ 0

Now consider (A-λI)^{j}X_{j} and (A-λI)^{j-1}X_{j}.