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Solution
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Solution provided by AtoZmath.com
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QR Decomposition (Gram Schmidt 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 (Gram Schmidt 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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`r_(11)=||q_1'||=sqrt(1^2+1^2+1^2+1^2)=sqrt(4)=2` | `q_1 = 1/(||q_1'||) * q_1'` | = | `1/2 * ` | | = | |
| `r_(12)=q_1^T * a_2` | = | | `xx` | | `=3` |
| `q_2'` | `=a_2-r_(12) * q_1` | = | | = | |
`r_(22)=||q_2'||=sqrt((-2.5)^2+2.5^2+2.5^2+(-2.5)^2)=sqrt(25)=5` | `q_2 = 1/(||q_2'||) * q_2'` | = | `1/5 * ` | | = | |
| `r_(13)=q_1^T * a_3` | = | | `xx` | | `=2` |
| `r_(23)=q_2^T * a_3` | = | | `xx` | | `=-2` |
| `q_3'` | `=a_3-r_(13) * q_1-r_(23) * q_2` | = | | = | |
`r_(33)=||q_3'||=sqrt(2^2+(-2)^2+2^2+(-2)^2)=sqrt(16)=4` | `q_3 = 1/(||q_3'||) * q_3'` | = | `1/4 * ` | | = | |
| `Q` | `=[q_1,q_2,q_3]` | = | | `0.5` | `-0.5` | `0.5` | | | `0.5` | `0.5` | `-0.5` | | | `0.5` | `0.5` | `0.5` | | | `0.5` | `-0.5` | `-0.5` | |
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| `R` | = | | `r_(11)` | `r_(12)` | `r_(13)` | | | `0` | `r_(22)` | `r_(23)` | | | `0` | `0` | `r_(33)` | |
| = | | `2` | `3` | `2` | | | `0` | `5` | `-2` | | | `0` | `0` | `4` | |
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checking `Q xx R = A?` | `Q xx R` | = | | `0.5` | `-0.5` | `0.5` | | | `0.5` | `0.5` | `-0.5` | | | `0.5` | `0.5` | `0.5` | | | `0.5` | `-0.5` | `-0.5` | |
| `xx` | | `2` | `3` | `2` | | | `0` | `5` | `-2` | | | `0` | `0` | `4` | |
| = | | `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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