Assignment 8 – MATH 2210Q

Assignment 8 – MATH 2210Q
Due April 21st, 2016
Problem 1) Assume that A is a 3 × 3 matrix with three distinct eigenvalues λ = 1, λ = 0.1 and λ = 0.5.
Let ~u be an eigenvector with associated eigenvalue λ = 1, ~v be an eigenvector with associated eigenvalue
λ = 0.1 and w
~ be an eigenvector with associated eigenvalue λ = 0.5.
(a) Explain in few words why {~u, ~v , w}
~ forms a basis of R3 .
(b) Denote by B the basis in (a). Assume that ~x0 ∈ R3 is such that [~x0 ]B = [2, 6, 7]. Define ~xk = A~xk−1
for k ≥ 0. Find
lim ~xk .
k→∞
Problem 2) Consider the matrix

3
2 1 1
 2
3 1 1
A=
 3
3 4 2
1 −1 0 2
We know that λ = 2 is an eigenvalue of A. Find the eigenspace




associated to λ = 2 and a basis for it.
Problem 3) Consider the matrix

2
 0
A=
 1
1
0
3
0
0
0
3
4
0

−1
4 

1 
1
(a) Find its characteristic polynomial. Use row or column expansions to find it.
(b) Find the eigenvalues of A.
Problem 4)Assume that A is a 3 × 3 matrix with three distinct eigenvalues λ = 1, λ = 2 and λ = 3.
Assume that ~u = [1, 1, 1] is an eigenvector with associated eigenvalue λ = 1, ~v = [1, 1, 0] is an eigenvector
with associated eigenvalue λ = 2 and w
~ = [0, 1, 2] is an eigenvector with associated eigenvalue λ = 3.
(a) Find an invertible matrix P and a diagonal matrix D such that A = P DP −1 .
(b) Compute P −1 .
(c) Using (a) and (b), find A100 .
1
2
− − − − − − − − − − − − − − − − − − − − − − − − −−
Also, you should look at the following problems in the textbook. These problems are not to be handed in.
Section 5.1: 1, 3, 5, 7, 9, 15, 17, 19, 21, 25, 31.
Section 5.2: 1, 3, 7, 9, 11, 15, 17, 21, 25.
Section 5.3: 1, 3, 5, 7, 11, 19, 21, 23, 25, 31.