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12. Cholesky Decomposition example
( Enter your problem )
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- Example `[[6,15,55],[15,55,225],[55,225,979]]`
- Example `[[6,-2,2],[-2,3,-1],[2,-1,3]]`
- Example `[[25,15,-5],[15,18,0],[-5,0,11]]`
- Example `[[8,-6,2],[-6,7,-4],[2,-4,3]]`
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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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1. Example `[[6,15,55],[15,55,225],[55,225,979]]`
Find Cholesky Decomposition ...
`[[6,15,55],[15,55,225],[55,225,979]]`
Solution:
Formula
`l_(ki)=(a_(ki) - sum_{j=1}^{i-1} l_(ij) * l_(kj))/(l_(ii))`
`l_(kk)=sqrt(a_(kk)-sum_{j=1}^{k-1} l_(kj)^2)`
Cholesky decomposition : `A=L*L^T`, Every symmetric positive definite matrix A can be decomposed into a product of a unique lower triangular matrix L and its transpose.
`l_(11)=sqrt(a_(11))=sqrt(6)=2.4495`
`l_(21)=(a_(21))/l_(11)=(15)/(2.4495)=6.1237`
`l_(22)=sqrt(a_(22)-l_(21)^2)=sqrt(55-(6.1237)^2)=sqrt(55-37.5)=4.1833`
`l_(31)=(a_(31))/l_(11)=(55)/(2.4495)=22.4537`
`l_(32)=(a_(32)-l_(31) xx l_(21))/l_(22)=(225-(22.4537)xx(6.1237))/(4.1833)=(225-137.5)/(4.1833)=20.9165`
`l_(33)=sqrt(a_(33)-l_(31)^2-l_(32)^2)=sqrt(979-(22.4537)^2-(20.9165)^2)=sqrt(979-941.6667)=6.1101`
| So `L` | = | | `l_(11)` | `0` | `0` | | | `l_(21)` | `l_(22)` | `0` | | | `l_(31)` | `l_(32)` | `l_(33)` | |
| = | | 2.4495 | 0 | 0 | | | 6.1237 | 4.1833 | 0 | | | 22.4537 | 20.9165 | 6.1101 | |
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| `L xx L^T` | = | | 2.4495 | 0 | 0 | | | 6.1237 | 4.1833 | 0 | | | 22.4537 | 20.9165 | 6.1101 | |
| `xx` | | 2.4495 | 6.1237 | 22.4537 | | | 0 | 4.1833 | 20.9165 | | | 0 | 0 | 6.1101 | |
| = | |
This material is intended as a summary. Use your textbook for detail explanation.
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