410 CHAPTER 14. ANALYSIS OF LINEAR TRANSFORMATIONS

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.

Hint: First verify that V is normal. Thus U∗VU = D. Now verify that D∗D = I.What does this mean for the diagonal entries of D? If you have a complex numberwhich has magnitude 1, what form does it take?

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 alge-braic condition ρ (A) < 1. Is it still the case that the Neumann series converges to(I−A)−1?