Page 402 - Applied Numerical Methods Using MATLAB

PROBLEMS 391
(a) Verify that the eigenvalues and eigenvectors of this matrix are as follows,
with N = 3 for convenience.
nπ

λ n = a + 2b cos for n = 1to N (P8.1.2)
N + 1

T
2 nπ 2nπ Nnπ
v n = sin sin ·· · sin (P8.1.3)
N + 1 N + 1 N + 1 N + 1
(b) Letting N = 3,a = 2, and b = 1, find the eigenvalues/eigenvectors of
the above matrix by using (P8.1.2,3) and by using the MATLAB routine
“eig_Jacobi()”or“eig()” for cross-check.
8.2 Circulant Matrix
Consider the following N × N circulant matrix as

h(0) h(N − 1) h(N − 2) ·· h(1)
 
 h(1) h(0) h(N − 1) ·· h(2) 
 
h(2) h(1) h(0) ··
 
 h(3)  (P8.2.1)
· · · ··
 
 · 
 · · · ·· · 
h(N − 1)h(N − 2) h(N − 3) ·· h(0)
(a) Vertify that the eigenvalues and eigenvectors of this matrix are as follows,
with N = 4 for convenience.
λ n = h(0) + h(N − 1)e j2πn/N + h(N − 2)e j2π2n/N (P8.2.2)
+· · · + h(1)e j2π(N−1)n/N
]
v n = [1 e j2πn/N e j2π2n/N ··· e j2π(N−1)n/N T (P8.2.3)
for n = 0to N − 1

(b) Letting N = 4,h(0) = 2,h(3) = h(1) = 1, and h(2) = 0, find the eigen-
values/eigenvectors of the above matrix by using (P8.2.2,3) and by using
the MATLAB routine “eig_Jacobi()”or “eig()”. Do they agree? Do
they satisfy Eq. (8.1.1)?
8.3 Solving a Vector Differential Equation by Decoupling: Diagonalization.
Consider the following two-dimensional vector differential equation (state
equation) as

x 1 (t) 0 1 x 1 (t) 0

= + u s (t) (P8.3.1)
x 2 (t) −2 −3 x 2 (t) 1

x 1 (0) 1
with = and u s (t) = 1 ∀ t ≥ 0
x 2 (0) 0