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9. Cholesky Decomposition example ( Enter your problem )
  1. Example-1
  2. Example-2
Other related methods
  1. Reduce matrix to Row Echelon Form
  2. Rank of matrix
  3. Characteristic polynomial of matrix
  4. Eigenvalues
  5. Eigenvectors
  6. Triangular Matrix
  7. LU Decomposition of matrix
  8. Diagonal Matrix
  9. Cholesky Decomposition
  10. QR Decomposition (Gram Schmidt Method)
  11. QR Decomposition (Householder Method)
  12. LQ Decomposition
  13. Pivots
  14. Singular Value Decomposition (SVD)
  15. Moore-Penrose Pseudoinverse
  16. Power Method for dominant eigenvalue
  17. determinants using Sarrus Rule
  18. determinants using properties of determinants

1. Example-1



1. 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.

Here `A` = 
61555
1555225
55225979


`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)`
 = 
2.449500
6.12374.18330
22.453720.91656.1101


`L xx L^T` = 
2.449500
6.12374.18330
22.453720.91656.1101
 `xx` 
2.44956.123722.4537
04.183320.9165
006.1101
 = 
61555
1555225
55225979


and `A` = 
61555
1555225
55225979



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