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15. LQ Decomposition example
( Enter your problem )
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- Example `[[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]`
- Example `[[3,-6],[4,-8],[0,1]]`
- Example `[[1,-4],[2,3],[2,2]]`
- Example `[[1,2,4],[0,0,5],[0,3,6]]`
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Other related methods
- Transforming matrix to Row Echelon Form
- Transforming matrix to Reduced Row Echelon Form
- Rank of matrix
- Characteristic polynomial of matrix
- Eigenvalues
- Eigenvectors (Eigenspace)
- Triangular Matrix
- LU decomposition using Gauss Elimination method of matrix
- LU decomposition using Doolittle's method of matrix
- LU decomposition using Crout's method of matrix
- Diagonal Matrix
- Cholesky Decomposition
- QR Decomposition (Gram Schmidt Method)
- QR Decomposition (Householder Method)
- LQ Decomposition
- Pivots
- Singular Value Decomposition (SVD)
- Moore-Penrose Pseudoinverse
- Power Method for dominant eigenvalue
- determinants using Sarrus Rule
- determinants using properties of determinants
- Row Space
- Column Space
- Null Space
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4. Example `[[1,2,4],[0,0,5],[0,3,6]]`
Find LQ Decomposition ...
`[[1,2,4],[0,0,5],[0,3,6]]`
Solution:
| Here `A` | = | | `1` | `2` | `4` | | | `0` | `0` | `5` | | | `0` | `3` | `6` | |
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Suppose you want a `LQ` factorization of a matrix `A`, then you do a QR factorization of `A^T`, i.e., `A^T=UR`, where `U` is orthogonal and `R` is upper triangular. Then `A=LQ=R^TU^T` where `L=R^T` is lower triangular, and `Q=U^T` is orthogonal.
| `A` | = | | `1` | `2` | `4` | | | `0` | `0` | `5` | | | `0` | `3` | `6` | |
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| `A'` | = | | `1` | `0` | `0` | | | `2` | `0` | `3` | | | `4` | `5` | `6` | |
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Now, UR Decomposition of `A'` by GramSchmidt Method
`r_(11)=||q_1'||=sqrt(1^2+2^2+4^2)=sqrt(21)=4.5826`
| `q_1 = 1/(||q_1'||) * q_1'` | = | `1/4.5826 * ` | | = | |
| `r_(12)=q_1^T * a_2` | = | | [ | `0.2182` | `0.4364` | `0.8729` | ] |
| `xx` | | `=4.3644` |
| `q_2'` | `=a_2-r_(12) * q_1` | = | | = | |
`r_(22)=||q_2'||=sqrt((-20/21)^2+(-40/21)^2+25/21^2)=sqrt(125/21)=2.4398`
| `q_2 = 1/(||q_2'||) * q_2'` | = | `1/2.4398 * ` | | = | | `-0.3904` | | | `-0.7807` | | | `0.488` | |
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| `r_(13)=q_1^T * a_3` | = | | [ | `0.2182` | `0.4364` | `0.8729` | ] |
| `xx` | | `=6.5465` |
| `r_(23)=q_2^T * a_3` | = | | [ | `-0.3904` | `-0.7807` | `0.488` | ] |
| `xx` | | `=0.5855` |
| `q_3'` | `=a_3-r_(13) * q_1-r_(23) * q_2` | = | | -6.5465 | | -0.5855 | | `-0.3904` | | | `-0.7807` | | | `0.488` | |
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| = | |
`r_(33)=||q_3'||=sqrt((-6/5)^2+3/5^2+0^2)=sqrt(9/5)=1.3416`
| `q_3 = 1/(||q_3'||) * q_3'` | = | `1/1.3416 * ` | | = | |
| `U` | `=[q_1,q_2,q_3]` | = | | `0.2182` | `-0.3904` | `-0.8944` | | | `0.4364` | `-0.7807` | `0.4472` | | | `0.8729` | `0.488` | `0` | |
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| `R` | = | | `r_(11)` | `r_(12)` | `r_(13)` | | | `0` | `r_(22)` | `r_(23)` | | | `0` | `0` | `r_(33)` | |
| = | | `4.5826` | `4.3644` | `6.5465` | | | `0` | `2.4398` | `0.5855` | | | `0` | `0` | `1.3416` | |
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Now, L and Q from R and U
| `L=R^T` | = | | `4.5826` | `0` | `0` | | | `4.3644` | `2.4398` | `0` | | | `6.5465` | `0.5855` | `1.3416` | |
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| `Q=U^T` | = | | `0.2182` | `0.4364` | `0.8729` | | | `-0.3904` | `-0.7807` | `0.488` | | | `-0.8944` | `0.4472` | `0` | |
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checking `L xx Q = A?`
| `L xx Q` | = | | `4.5826` | `0` | `0` | | | `4.3644` | `2.4398` | `0` | | | `6.5465` | `0.5855` | `1.3416` | |
| `xx` | | `0.2182` | `0.4364` | `0.8729` | | | `-0.3904` | `-0.7807` | `0.488` | | | `-0.8944` | `0.4472` | `0` | |
| = | | `1` | `2` | `4` | | | `0` | `0` | `5` | | | `0` | `3` | `6` | |
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| and `A` | = | | `1` | `2` | `4` | | | `0` | `0` | `5` | | | `0` | `3` | `6` | |
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This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
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