MATH 235: Linear Algebra 2 - Fall 2022 - Written Assignment 12
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MATH 235: Linear Algebra 2
Fall 2022
Written Assignment 12
Q1. Consider A = ┌0(1)
'1
1(1) ┐
一1'.
(a) Find a singular value decomposition for A. Justify your answer.
(b) Find the maximum and minimum of |Ayx| subject to |yx| = 2. Find vectors at which
the maximum and minimum occurs. Justify your answer.
Q2. Consider
|
┌ 1(1) A = '(')1 '1 |
1 一1 1 一1 |
1 1 一1 一1 |
|
0(0) 0(0)┐' ┌' 1一 |
一1/2 1/2 一1/10 一7/10 |
一1/10 一1/10 49/50 一7/50 |
(Each of the three matrices in the above product has orthogonal columns. right most matrix have unit length.)
一(一)
┐'
一1
'' .
All columns of the
(a) Find rank(A) and nullity(AT A).
(b) Find a basis for Col(A), Row(A), Null(A), and Null(AT) (see Video 4 of Week 12).
|
Q3. Find a singular value decomposition of A = - 2(1) |
1 2 |
1 2 |
2(1)_. |
Q4. Let A ∈ M3|3(R) such that λ = (1 + 3i) is a complex eigenvalue of A and det(A) = 20. Prove
that λ = 1 一 3i is also an eigenvalue of A. What is the third eigenvalue of A?
2022-12-05