Lecture Notes in Differential Equations - Bruce E. Shapiro

321
Corollary 31.12. The generalized eigenvectors of the m × m matrix
⎛
⎞
λ 1 0 0
.
B =
0 λ .. .
⎜
⎝
.
. .. . ⎟
(31.140)
.. 1 ⎠
0 · · · 0 λ
are the standard basis vectors e 1 , e 2 , ..., e m .
Proof.
(B − λI)e 1 = λe 1 − λe 1 = 0 (31.141)
(B − λI)e 2 = Be 2 − λe 2 (31.142)
= (1, λ, 0, . . . , 0) T − (0, λ, 0, . . . , 0, ) T = e 1 (31.143)
(B − λI)e 3 = (0, 1, λ, 0 . . . , 0) T − (0, 0, λ, 0, . . . , 0) T = e 2 (31.144)
.
(B − λI)e m = (0, . . . , 0, 1, λ)) T − (0, . . . , 0, 1, 0) T = e m−1 (31.145)
Returning to equation (31.128), which gave the solution z of the ith Jordan
Block of the Jordan form z ′ = Jz of the differential equation, we will now
make the transformation back to the space of the usual y variable, using
the fact that y = Uz
y = Uz (31.146)
= a 1 e λt w 1 + a 2 e λt (w 2 + w 1 t)
(
+ · · · + a m e λt t m−1 )
w m + w m−1 t + · · · + w 1
(m − 1)!
(31.147)
where
w i = Ue i (31.148)
are the generalized eigenvectors of A. This establishes the following theorem.