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3. Let H be as in exercise (1). Let x be a vector such that Hx = α€₁. Show that • α = ±||x||2. • He₁ = x/a. x¹ Hej = 0 for j = 1 You may assume that the results of exercise (1) are correct.

3. Let H be as in exercise (1). Let x be a vector such that Hx = α€₁. Show that • α = ±||x||2. • He₁ = x/a. x¹ Hej = 0 for j = 1 You may assume that the results of exercise (1) are correct.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Based on the information in Question 1, solve Question 3

1. Show that the matrix satisfies H=1-2ww¹, ||w||2 = 1 • H = HT • HTH = H² = I. The matrix H is called a Householder transformation.
Transcribed Image Text:1. Show that the matrix satisfies H=1-2ww¹, ||w||2 = 1 • H = HT • HTH = H² = I. The matrix H is called a Householder transformation.
3. Let H be as in exercise (1). Let x be a vector such that Hx = α€₁. Show that • α = ±||x||2. • He₁ = x/a. x¹ Hej = 0 for j #1 You may assume that the results of exercise (1) are correct.
Transcribed Image Text:3. Let H be as in exercise (1). Let x be a vector such that Hx = α€₁. Show that • α = ±||x||2. • He₁ = x/a. x¹ Hej = 0 for j #1 You may assume that the results of exercise (1) are correct.
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