Lecture Notes in Differential Equations - Bruce E. Shapiro

320 LESSON 31. LINEAR SYSTEMS
(
z = a 1 e λt e 1 + a 2 e λt (e 2 + e 1 t) + a 3 e λt e 3 + e 2 t + 1 )
2 e 1t 2 +
(
+ · · · + a m e λt t m−1 )
e m + e m−1 t + · · · + e 1
(31.128)
(m − 1)!
Definition 31.11. Let (λ, v) be an eigenvalue-eigenvector pair ofA with
multiplicity m. Then the set of generalized eigenvectors of A corresponding
to the eigenvalue λ are the vectors w 1 , ..., w m where
For k = 1, equation (31.129) gives
i.e.,
(A − λI) k w k = 0, k = 1, 2, ..., m (31.129)
0 = (A − λI)w 1 = Aw 1 − λw 1 (31.130)
w 1 = v (31.131)
So one of the generalized eigenvectors corresponding to the eigenvector v is
always the original eigenvector v itself. If m = 1, this is the only generalized
eigenvector. If m > 1, there are additional generalized eigenvectors.
For k = 2, equation (31.129) gives
Rearranging,
0 = (A − λI) 2 w 2 (31.132)
= (A − λI)(A − λI)w 2 (31.133)
= A(A − λI)w 2 − λ(A − λI)w 2 (31.134)
A(A − λI)w 2 = λ(A − λI)w 2 (31.135)
Thus (A − λI)w 2 is an eigenvector of A with eigenvalue λ. Since w 1 = v is
also an eigenvector with eigenvalue λ, we call it a generalized eigenvector.
Thus we also have, from equation 31.135 and 31.129
(A − λI)w 2 = w 1 (31.136)
In general, if the multiplicity of the eigenvalue is m,
(A − λI)w 1 = 0 (31.137)
(A − λI)w 2 = w 1 (31.138)
.
.
(A − λI)w m = w m−1 (31.139)