Math 477/577 — Computer Assignment 5, due Nov.9, 2006
1. Let Abe a 10×10 random matrix with entries from the standard normal distribution, minus twice
the identity. Write a program to plot ketAk2against tfor 0 ≤t≤20 on a log scale, comparing
the result to the straight line etα(A), where α(A) = maxj<(λj) is the spectral abscissa of A. Run
the program for ten random matrices Aand comment on the results. What property of a matrix
leads to a ketAk2curve that remains oscillatory as t→ ∞?
Hints: Recall that the matrix exponential is defined as
etA =
∞
X
k=0
Aktk
k!.
Use randn and expm to deal with normally distributed random numbers and matrix exponentials,
respectively.
2. Let Abe the 32 ×32 matrix with −1 on the main diagonal, 1 on the first and second superdiag-
onals, and 0 elsewhere.
For A∈Cm×mwith spectrum Λ(A)⊆Cand ε > 0, we define the 2-norm ε-pseudospectrum of
A, Λε(A), to be the set of numbers z∈Csatisfying any of the following conditions:
(i) zis an eigenvalue of A+δA for some δA with kδAk2≤ε;
(ii) There exists a vector u∈Cmwith k(A−zI)uk2≤εand kuk2= 1;
(iii) σm(zI −A)≤ε;
(iv) k(zI −A)−1k2≥ε−1.
The matrix (zI −A)−1in (iv) is known as the resolvent of Aat z; if zis an eigenvalue of A, we
use the convention k(zI −A)−1k2=∞. In (iii), σmdenotes the smallest singular value of A.
(a) Using an SVD algorithm built into MATLAB together with MATLAB’s contour com-
mand, generate a plot of the boundaries of the 2-norm ε-pseudospectra of Afor ε=
10−1,10−2, . . . , 10−8.
(b) Produce a semilogy plot of ketAk2against tfor 0 ≤t≤50. What is the initial growth
rate of the curve before the eventual decay sets in? Can you relate this to your plot of
pseudospectra? (Compare to the previous problem.)