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Please report violations to [email protected] eAt = P .. .J
e P−1 with eJ t .. t . ⎜
⎜
⎜
⎜
⎜
⎜
=⎜
⎜
⎜
⎜
⎜
⎝ teλt te
2! eλt eλt teλt
.. ··· t
e
(k − 1)! .. .
.
. .. . . . eλt t2 eλt
2!
teλt D
E eλt while setting f (z ) = e zt in (7.9.9) produces
s e At ki −1 j λi t =
i=1 j =0 te
(A − λi I)j Gi .
j! ⎟
⎟
⎟
⎟
⎟
⎟
⎟ , (7.9.18)
⎟
⎟
⎟
⎟
⎠ T
H (7.9.19) Either of these can be used to show that the three properties (7.4.3)–(7.4.5)
on p. 541 still hold. In particular, d eAt /dt = AeAt = eAt A, so, just as in
the diagonalizable case, u(t) = eAt c is the unique solution of (7.9.17) (the
uniqueness argument given in §7.4 remains valid). In the diagonalizable case,
the solution of (7.9.17) involves only the eigenvalues and eigenvectors of A as
described in (7.4.7) on p. 542, but generalized eigenvectors are needed for the
nondiagonalizable case. Using (7.9.19) yields the solution to (7.9.17) as
s u(t) = eAt c = ki −1 j λi t IG
R te
vj (λi ), where vj (λi ) = (A − λi I)j Gi c. (7.9.20)
j! Y
P i=1 j =0 Each vki −1 (λi ) is an eigenvector associated with λi because (A − λi I)ki Gi = 0,
and {vki −2 (λi ), . . . , v1 (λi ), v0 (λi )} is an associated chain of generalized eigenvectors. The behavior of the solution (7.9.20) as t → ∞ is similar but not
identical to that discussed on p. 544 because for λ = x + iy and t > 0,
⎧
if x < 0,
⎪0
⎪
⎪ unbounded
⎨
if x ≥ 0 and j > 0,
tj eλt = tj ext (cos yt + i sin yt) → oscillates indeﬁnitely if x = j = 0 and y = 0,
⎪
⎪1
⎪
if x = y = j = 0.
⎩ O
C In particular, if Re (λi ) < 0 for every λi ∈ σ (A) , then u(t) → 0 for every
initial vector c, in which case the system is said to be stable .
• Nonhomogeneous Systems. It can be veriﬁed by direct manipulation
that the solution of u (t) = Au(t) + f (t) with u(t0 ) = c is given by
u(t) = eA(t−t0 ) c + t eA(t−τ ) f (τ )dτ...

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