14.5. EXERCISES 409

where D is the diagonal matrix obtained from the eigenvalues of A and Nε is a nilpo-tent matrix commuting with D which is very small provided ε is chosen very small.Now let Ψ(t) be the solution of

Ψ′ =−DΨ, Ψ(0) = I

described earlier as∞

∑k=0

(−1)k tkDk

k!.

Thus Ψ(t) commutes with D and Nε . Tell why. Next argue

(Ψ(t)z)′ = Ψ(t)Nεz (t)

and integrate from 0 to t. Then

Ψ(t)z (t)−z0 =∫ t

0Ψ(s)Nεz (s)ds.

It follows||Ψ(t)z (t)|| ≤ ||z0||+

∫ t

0||Nε || ||Ψ(s)z (s)||ds.

It follows from Gronwall’s inequality

||Ψ(t)z (t)|| ≤ ||z0||e||Nε ||t

Now look closely at the form of Ψ(t) to get an estimate which is interesting. Explainwhy

Ψ(t) =

eµ1t 0

. . .

0 eµnt

and now observe that if ε is chosen small enough, ||Nε || is so small that each com-ponent of z (t) converges to 0.

25. Using Problem 24 show that if A is a matrix having the real parts of all eigenvaluesless than 0 then if

Ψ′ (t) = AΨ(t) , Ψ(0) = I

it followslimt→∞

Ψ(t) = 0.

Hint: Consider the columns of Ψ(t)?

26. Let Ψ(t) be a fundamental matrix satisfying

Ψ′ (t) = AΨ(t) , Ψ(0) = I.

Show Ψ(t)n = Ψ(nt) . Hint: Subtract and show the difference satisfies Φ′ = AΦ,and Φ(0) = 0. Use uniqueness.