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Solution
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Solution provided by AtoZmath.com
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QR Decomposition (Householder Method) calculator
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 1. `[[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]`
2. `[[1,-4],[2,3],[2,2]]`
3. `[[3,-6],[4,-8],[0,1]]`
4. `[[1,2,4],[0,0,5],[0,3,6]]`
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Example1. 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` | |
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| `A_1` | = | | `1` | `-1` | `4` | | | `1` | `4` | `-2` | | | `1` | `4` | `2` | | | `1` | `-1` | `0` | |
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`||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` | = | | + | 2 | `xx` | | = | |
| `H_1=I-2*(v_1*v_1^T)/(v_1^T*v_1)` | = | | `-` | `2/12` | `*` | | `*` | | = | | `-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` | |
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| `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` | |
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Now removing 1st row and 1st column, we get | `A_2` | = | | `10/3` | `-4` | | | `10/3` | `0` | | | `-5/3` | `-2` | |
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`||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` | = | | + | 5 | `xx` | | = | |
| `H_2=I-2*(v_2*v_2^T)/(v_2^T*v_2)` | = | | `-` | `2/250/3` | `*` | | `*` | | = | | `-2/3` | `-2/3` | `1/3` | | | `-2/3` | `11/15` | `2/15` | | | `1/3` | `2/15` | `14/15` | |
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| `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` | |
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Now removing 1st row and 1st column, we get `||a_3||=sqrt(12/5^2+(-16/5)^2)=sqrt(16)=4` | `v_3=a_3+sign(a_(11))||a_3||e_1` | = | | + | 4 | `xx` | | = | |
| `H_3=I-2*(v_3*v_3^T)/(v_3^T*v_3)` | = | | `-` | `2/256/5` | `*` | | `*` | | = | |
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` | |
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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` | |
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| and `A` | = | | `1` | `-1` | `4` | | | `1` | `4` | `-2` | | | `1` | `4` | `2` | | | `1` | `-1` | `0` | |
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