We use cookies to improve your experience on our site and to show you relevant advertising. By browsing this website, you agree to our use of cookies. Learn more   

AtoZmath.com
Support us
(New) All problem can be solved using search box
I want to sell my website www.AtoZmath.com with complete code
Home What's new College Algebra Games Feedback About us
Algebra Matrix & Vector Numerical Methods Statistical Methods Operation Research Word Problems Calculus Geometry Pre-Algebra


14. QR Decomposition (Householder Method) example ( Enter your problem )
( Enter your problem )
  1. Example `[[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]`
  2. Example `[[2,-2,18],[2,1,0],[1,2,0]]`
  3. Example `[[1,-4],[2,3],[2,2]]`
  4. Example `[[1,2,4],[0,0,5],[0,3,6]]`
 		Other related methods
  1. Transforming matrix to Row Echelon Form
  2. Transforming matrix to Reduced Row Echelon Form
  3. Rank of matrix
  4. Characteristic polynomial of matrix
  5. Eigenvalues
  6. Eigenvectors (Eigenspace)
  7. Triangular Matrix
  8. LU decomposition using Gauss Elimination method of matrix
  9. LU decomposition using Doolittle's method of matrix
  10. LU decomposition using Crout's method of matrix
  11. Diagonal Matrix
  12. Cholesky Decomposition
  13. QR Decomposition (Gram Schmidt Method)
  14. QR Decomposition (Householder Method)
  15. LQ Decomposition
  16. Pivots
  17. Singular Value Decomposition (SVD)
  18. Moore-Penrose Pseudoinverse
  19. Power Method for dominant eigenvalue
  20. determinants using Sarrus Rule
  21. determinants using properties of determinants
  22. Row Space
  23. Column Space
  24. Null Space
13. QR Decomposition (Gram Schmidt Method)
(Previous method)		2. Example `[[2,-2,18],[2,1,0],[1,2,0]]`
(Next example)

1. Example `[[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]`


Find QR Decomposition (Householder Method) ...
`[[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]`

Solution:
Here `A` = 
`1``-1``4`
`1``4``-2`
`1``4``2`
`1``-1``0`


`A_1` = 
`1``-1``4`
`1``4``-2`
`1``4``2`
`1``-1``0`


`a_1` = 
`1`
`1`
`1`
`1`


`||a_1||=sqrt(1^2+1^2+1^2+1^2)=sqrt(4)=2`

`v_1=a_1+sign(a_(11))||a_1||e_1` = 
`1`
`1`
`1`
`1`
 + 2 `xx` 
`1`
`0`
`0`
`0`
 = 
`3`
`1`
`1`
`1`


`H_1=I-2*(v_1*v_1^T)/(v_1^T*v_1)` = 
1000
0100
0010
0001
`-``2/12``*`
`3`
`1`
`1`
`1`
`*`
[`3``1``1``1`]
 = 
`-1/2``-1/2``-1/2``-1/2`
`-1/2``5/6``-1/6``-1/6`
`-1/2``-1/6``5/6``-1/6`
`-1/2``-1/6``-1/6``5/6`


`H_1 * A_1` = 
`-1/2``-1/2``-1/2``-1/2`
`-1/2``5/6``-1/6``-1/6`
`-1/2``-1/6``5/6``-1/6`
`-1/2``-1/6``-1/6``5/6`
 `xx` 
`1``-1``4`
`1``4``-2`
`1``4``2`
`1``-1``0`
 = 
`-2``-3``-2`
`0``10/3``-4`
`0``10/3``0`
`0``-5/3``-2`



Now removing 1st row and 1st column, we get
`A_2` = 
`10/3``-4`
`10/3``0`
`-5/3``-2`


`a_2` = 
`10/3`
`10/3`
`-5/3`


`||a_2||=sqrt(10/3^2+10/3^2+(-5/3)^2)=sqrt(25)=5`

`v_2=a_2+sign(a_(11))||a_2||e_1` = 
`10/3`
`10/3`
`-5/3`
 + 5 `xx` 
`1`
`0`
`0`
 = 
`25/3`
`10/3`
`-5/3`


`H_2=I-2*(v_2*v_2^T)/(v_2^T*v_2)` = 
100
010
001
`-``2/250/3``*`
`25/3`
`10/3`
`-5/3`
`*`
[`25/3``10/3``-5/3`]
 = 
`-2/3``-2/3``1/3`
`-2/3``11/15``2/15`
`1/3``2/15``14/15`


`H_2 * A_2` = 
`-2/3``-2/3``1/3`
`-2/3``11/15``2/15`
`1/3``2/15``14/15`
 `xx` 
`10/3``-4`
`10/3``0`
`-5/3``-2`
 = 
`-5``2`
`0``12/5`
`0``-16/5`



Now removing 1st row and 1st column, we get
`A_3` = 
`12/5`
`-16/5`


`a_3` = 
`12/5`
`-16/5`


`||a_3||=sqrt(12/5^2+(-16/5)^2)=sqrt(16)=4`

`v_3=a_3+sign(a_(11))||a_3||e_1` = 
`12/5`
`-16/5`
 + 4 `xx` 
`1`
`0`
 = 
`32/5`
`-16/5`


`H_3=I-2*(v_3*v_3^T)/(v_3^T*v_3)` = 
10
01
`-``2/256/5``*`
`32/5`
`-16/5`
`*`
[`32/5``-16/5`]
 = 
`-3/5``4/5`
`4/5``3/5`


`H_3 * A_3` = 
`-3/5``4/5`
`4/5``3/5`
 `xx` 
`12/5`
`-16/5`
 = 
`-4`
`0`



Since, `H_3H_2H_1A=R`

`H_3H_2H_1A=`
`1``0``0``0`
`0``1``0``0`
`0``0``-3/5``4/5`
`0``0``4/5``3/5`
 `xx` 
`1``0``0``0`
`0``-2/3``-2/3``1/3`
`0``-2/3``11/15``2/15`
`0``1/3``2/15``14/15`
 `xx` 
`-1/2``-1/2``-1/2``-1/2`
`-1/2``5/6``-1/6``-1/6`
`-1/2``-1/6``5/6``-1/6`
`-1/2``-1/6``-1/6``5/6`
 `xx` 
`1``-1``4`
`1``4``-2`
`1``4``2`
`1``-1``0`
 = 
`-2``-3``-2`
`0``-5``2`
`0``0``-4`
`0``0``0`
 = R


Also `A=H_1H_2H_3R` and `A=QR`, `:.Q=H_1H_2H_3`

`Q=H_1H_2H_3`=
`-1/2``-1/2``-1/2``-1/2`
`-1/2``5/6``-1/6``-1/6`
`-1/2``-1/6``5/6``-1/6`
`-1/2``-1/6``-1/6``5/6`
 `xx` 
`1``0``0``0`
`0``-2/3``-2/3``1/3`
`0``-2/3``11/15``2/15`
`0``1/3``2/15``14/15`
 `xx` 
`1``0``0``0`
`0``1``0``0`
`0``0``-3/5``4/5`
`0``0``4/5``3/5`
 = 
`-1/2``1/2``-1/2``-1/2`
`-1/2``-1/2``1/2``-1/2`
`-1/2``-1/2``-1/2``1/2`
`-1/2``1/2``1/2``1/2`



checking `Q xx R = A?`

`Q xx R` = 
`-1/2``1/2``-1/2``-1/2`
`-1/2``-1/2``1/2``-1/2`
`-1/2``-1/2``-1/2``1/2`
`-1/2``1/2``1/2``1/2`
 `xx` 
`-2``-3``-2`
`0``-5``2`
`0``0``-4`
`0``0``0`
 = 
`1``-1``4`
`1``4``-2`
`1``4``2`
`1``-1``0`


and `A` = 
`1``-1``4`
`1``4``-2`
`1``4``2`
`1``-1``0`


This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then Submit Here


13. QR Decomposition (Gram Schmidt Method)
(Previous method)		2. Example `[[2,-2,18],[2,1,0],[1,2,0]]`
(Next example)


Share this solution or page with your friends.


Home What's new College Algebra Games Feedback About us
 
Copyright © 2023. All rights reserved. Terms, Privacy
 
 

.