Explanation Given: A = [ 1 − 5 / 3 2 12 2 8 4 15 ] . Approach: Let u be a nonzero vector in R n and let I be the ( n × n ) identity matrix. The ( n × n ) matrix S given by, S = I − 2 u T u u u T is called a Householder matrix. Calculation: Consider a Householder matrix Q 1 such that the last three component in Q 1 A 1 is zero. Here, A 1 = [ 1 , 2 , 2 , 4 ] T . Now, Q 1 = I − b u u T and b = 2 u T u , u i = 0 , for i = 1 , ... , k − 1 and u k = v k − s , Here, s = ± v k 2 + v k + 1 2 + ... + v n 2 , and u i = v i for i = k + 1 , ... , n For k = 1 s = ± v 1 2 + v 2 2 + v 3 2 + v 4 2 = ± 1 2 + 2 2 + 2 2 + 4 2 = ± 5 Choose s = − 5 so that s v 1 ≤ 0 Therefore, u 1 = [ 6 , 2 , 2 , 4 ] T that is, u 1 = [ 6 2 2 4 ] So, u 1 T u 1 = [ 6 2 2 4 ] [ 6 2 2 4 ] = 36 + 4 + 4 + 16 = 60 u 1 T A 1 = [ 6 2 2 4 ] [ 1 2 2 4 ] = 6 + 4 + 4 + 16 = 30 u 1 T A 2 = [ 6 2 2 4 ] [ − 5 3 12 8 15 ] = − 10 + 24 + 16 + 60 = 90 b = 2 60 = 1 30 Q 1 A 1 = ( I − b u 1 u 1 T ) A 1 = A 1 − u 1 T A 1 30 u 1 = A 1 − 30 30 u 1 = A 1 − u 1 Q 1 A 2 = ( I − b u 1 u 1 T ) A 2 = A 2 − u 1 T A 2 30 u 1 = A 2 − ( 90 ) 30 u 1 = A 2 − 3 u 1 Now, Q 1 A = [ Q 1 A 1 , Q 1 A 2 ] Q 1 A = [ ( A 1 − u 1 ) ( A 2 − 3 u 1 ) ] = [ [ 1 2 2 4 ] − [ 6 2 2 4 ] [ − 5 3 12 8 15 ] − 3 [ 6 2 2 4 ] ] = [ − 5 − 59 3 0 6 0 2 0 3 ] The matrix [ − 5 − 59 3 0 6 0 2 0 3 ] is not in trapezoid form

Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

5th Edition

ISBN: 9780134689531

Author: Lee Johnson, Dean Riess, Jimmy Arnold

Publisher: PEARSON

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Chapter 7.6, Problem 11E

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The given matrix A=[1−5/321228415] into upper-trapezoidal form using Household matrices.

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Chapter 7.6, Problem 10E

Chapter 7.6, Problem 12E

Chapter 7 Solutions

Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

Ch. 7.1 - In Exercises 16, find a symmetric matrix A such...Ch. 7.1 - Prob. 2E Ch. 7.1 - In Exercises 16, find a symmetric matrix A such...Ch. 7.1 - Prob. 4E Ch. 7.1 - In Exercises 16, find a symmetric matrix A such...Ch. 7.1 - Prob. 6E Ch. 7.1 - In Exercises 712, find a substitution x=Qy that...Ch. 7.1 - In Exercises 712, find a substitution x=Qy that...Ch. 7.1 - In Exercises 712, find a substitution x=Qy that...Ch. 7.1 - In Exercises 712, find a substitution x=Qy that...

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The given matrix A = [ 1 − 5 / 3 2 12 2 8 4 15 ] into upper-trapezoidal form using Household matrices.

Solution Summary: The author explains that the upper-trapezoidal form of the given matrix is, left[cc1& -5/3 2& 122& 8 4

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To find:

The given matrix A=[1−5/321228415] into upper-trapezoidal form using Household matrices.

Want to see the full answer?

Chapter 7 Solutions

The given matrix A = [ 1 − 5 / 3 2 12 2 8 4 15 ] into upper-trapezoidal form using Household matrices.

Solution Summary: The author explains that the upper-trapezoidal form of the given matrix is, left[cc1& -5/3 2& 122& 8 4

Videos

To find: The given matrix A=[1−5/321228415] into upper-trapezoidal form using Household matrices.

Want to see the full answer?

Chapter 7 Solutions

To find:

The given matrix A=[1−5/321228415] into upper-trapezoidal form using Household matrices.