Math 1080 Homework #8

Math 1080
Homework #8 
Problem 1:
Find the diagonalization 𝐴 = 𝑋Λ𝑋
−1 of the following matrix:
𝐴 = [
3 0 0 0
0 −1 3 1
−2 0 4 0
2 −2 1 2
]
Problem 2:
Find the Schur factorization 𝐴 = 𝑄𝑇𝑄
𝑇
for the following matrix.
(Hint: Follow the proof of existence of the factorization.)
𝐴 = [
4 −2 1
−2 4 2
1 1 4
]
Problem 3:
Calculate the Rayleigh quotients 𝑟𝑘 = 𝑟(𝑥𝑘
) for the following matrix 𝐴 and given vectors 𝑥𝑘. How far is each
𝑟𝑘 from the closest eigenvalue of 𝐴?
𝐴 = [
4 6 1
6 4 6
1 6 4
],
𝑥1 = [
1.5
2
1
], 𝑥2 = [
1
2.1
1
], 𝑥3 = [
1
0
−1.1
], 𝑥4 = [
1
1
1
]
Problem 4:
Let 𝐴 be a symmetric matrix and let 𝜆1 ≤ 𝜆2 ≤ ⋯ ≤ 𝜆𝑛 be its eigenvalues. Show that for any 𝑥 ≠ 0 the
Rayleigh quotient 𝑟(𝑥) =
𝑥
𝑇𝐴𝑥
𝑥
𝑇𝑥
obeys 𝜆1 = min
x≠0
𝑟(𝑥) and 𝜆𝑛 = max
x≠0
𝑟(𝑥).
(Hint: Use orthogonal diagonalization of the matrix A.)