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Home > Matrix & Vector calculators > QR Decomposition (Householder Method) example
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]]`


1. 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)


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