14.5. EXERCISES 405

14. A matrix A is diagonally dominant if |aii| > ∑ j ̸=i∣∣ai j∣∣ . Show that the Gauss Seidel

method converges if A is diagonally dominant.

15. Suppose f (λ ) = ∑∞n=0 anλ

n converges if |λ |< R. Show that if ρ (A)< R where A isan n×n matrix, then

f (A)≡∞

∑n=0

anAn

converges in L (Fn,Fn) . Hint: Use Gelfand’s theorem and the root test.

16. Referring to Corollary 14.3.5, for λ = a+ ib show

exp(λ t) = eat (cos(bt)+ isin(bt)) .

Hint: Let y(t) = exp(λ t) 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) = 0v′′+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)+ isin(bt) and so y(t) = eat (cos(bt)+ isin(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 ′′+b2 f = 0

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

ddt

(( f ′)2

2+b2 f 2

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=0

tk(−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.