356 CHAPTER 13. NORMS

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 componentof 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, Φ (0) = 0.

Use uniqueness.

27. If the real parts of the eigenvalues of A are all negative, show that for every positivet,

limn→∞

Ψ(nt) = 0.

Hint: Pick Re (σ (A)) < −λ < 0 and use Problem 18 about the spectrum of Ψ (t)and Gelfand’s theorem for the spectral radius along with Problem 26 to argue that∣∣∣∣Ψ(nt) /e−λnt

∣∣∣∣ < 1 for all n large enough.

28. Let H be a Hermitian matrix. (H = H∗) . Show that eiH ≡∑∞

n=0(iH)n

n! is unitary.

29. Show the converse of the above exercise. If V is unitary, then V = eiH for some HHermitian.

30. If U is unitary and does not have −1 as an eigenvalue so that (I + U)−1

exists, showthat

H = i (I − U) (I + U)−1

is Hermitian. Then, verify that

U = (I + iH) (I − iH)−1.

31. Suppose that A ∈ L (V, V ) where V is a normed linear space. Also suppose that∥A∥ < 1 where this refers to the operator norm on A. Verify that

(I −A)−1

=

∞∑i=0

Ai

This is called the Neumann series. Suppose now that you only know the algebraiccondition ρ (A) < 1. Is it still the case that the Neumann series converges to (I −A)

−1?