352 CHAPTER 13. NORMS

16. Referring to Corollary 13.4.4, for λ = a+ ib show

exp (λt) = eat (cos (bt) + i sin (bt)) .

Hint: Let y (t) = exp (λt) and let z (t) = e−aty (t) . Show

z′′ + b2z = 0, z (0) = 1, z′ (0) = ib.

Now letting z = u+ iv where u, v are real valued, show

u′′ + b2u = 0, u (0) = 1, u′ (0) = 0

v′′ + b2v = 0, v (0) = 0, v′ (0) = b.

Next show u (t) = cos (bt) and v (t) = sin (bt) work in the above and that there is atmost one solution to

w′′ + b2w = 0 w (0) = α,w′ (0) = β.

Thus z (t) = cos (bt) + i sin (bt) and so y (t) = eat (cos (bt) + i sin (bt)). To show thereis at most one solution to the above problem, suppose you have two, w1, w2. Subtractthem. Let f = w1 − w2. Thus

f ′′ + b2f = 0

and f is real valued. Multiply both sides by f ′ and conclude

d

dt

((f ′)

2

2+ b2

f2

2

)= 0

Thus the expression in parenthesis is constant. Explain why this constant must equal0.

17. Let A ∈ L (Rn,Rn) . Show the following power series converges in L (Rn,Rn).

Ψ (t) ≡∞∑k=0

tkAk

k!

This was done in the chapter. Go over it and be sure you understand it. This ishow you can define exp (tA). Next show that Ψ′ (t) = AΨ(t) ,Ψ(0) = I. Next let

Φ (t) =∑∞

k=0tk(−A)k

k! . Show each Φ (t) ,Ψ(t) each commute with A. Next show thatΦ (t)Ψ (t) = I for all t. Finally, solve the initial value problem

x′ = Ax+ f , x (0) = x0

in terms of Φ and Ψ. This yields most of the substance of a typical differentialequations course.

18. In Problem 17 Ψ (t) is defined by the given series. Denote by exp (tσ (A)) the numbersexp (tλ) where λ ∈ σ (A) . Show exp (tσ (A)) = σ (Ψ (t)) . This is like Lemma 13.4.8.Letting J be the Jordan canonical form for A, explain why

Ψ (t) ≡∞∑k=0

tkAk

k!= S

∞∑k=0

tkJk

k!S−1