Math 1080: Homework #6

Description

Problem 1: Compute LU factorization of the matrix
𝐴 = [
3 0 −2 −2
0 −1 3 −1
−2 0 3 0
2 −2 1 2
]
Problem 2:
Solve the following system of equations by both LU factorization and QR factorization:
2𝑥1 − 𝑥3 = −7
2𝑥1 + 2𝑥2 + 3𝑥3 = 1
𝑥1 + 𝑥2 + 3𝑥3 = 2
Problem 3:
Let 𝐴 be nonsingular 𝑛 × 𝑛 matrix. Show that 𝐴 has LU factorization 𝐴 = 𝐿𝑈 (no
pivoting) with the diagonal terms of the matrix 𝑈 all nonzero if and only if for each 1 ≤
𝑘 ≤ 𝑛 the upper left 𝑘 × 𝑘 submatrix 𝐴1:𝑘,1:𝑘 is nonsingular.
(Hint: Use induction argument).
Problem 4:
Compute the LU factorization with partial pivoting, (i.e., find P, L, U such that
PA = LU
) for the following matrix
𝐴 = [
−1 −2 −1 −2
3 −1 1 −1
3 0 2 −1
0 1 −1 0
]