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Find cholesky decomposition [[8,-6,2],[-6,7,-4],[2,-4,3]] [ Calculator, Method and examples ]

Solution:
Your problem `->` cholesky decomposition [[8,-6,2],[-6,7,-4],[2,-4,3]]


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 matrix is symmetric positive definate, so Cholesky decomposition is possible.

A matrix is positive definite if it’s symmetric and all its pivots are positive.

`A` = 
`8``-6``2`
`-6``7``-4`
`2``-4``3`


Test method 1: Existence of all Positive Pivots.
First apply Gaussian Elimination method to find Pivots
`A` = 
`8``-6``2`
`-6``7``-4`
`2``-4``3`


`R_2 larr R_2+3/4xx R_1`

 = 
`8``-6``2`
 `0` `0=-6+3/4xx8`
`R_2 larr R_2+3/4xx R_1`
 `5/2` `5/2=7+3/4xx-6`
`R_2 larr R_2+3/4xx R_1`
 `-5/2` `-5/2=-4+3/4xx2`
`R_2 larr R_2+3/4xx R_1`
`2``-4``3`


`R_3 larr R_3-1/4xx R_1`

 = 
`8``-6``2`
`0``5/2``-5/2`
 `0` `0=2-1/4xx8`
`R_3 larr R_3-1/4xx R_1`
 `-5/2` `-5/2=-4-1/4xx-6`
`R_3 larr R_3-1/4xx R_1`
 `5/2` `5/2=3-1/4xx2`
`R_3 larr R_3-1/4xx R_1`


`R_3 larr R_3+ R_2`

 = 
`8``-6``2`
`0``5/2``-5/2`
 `0` `0=0+0`
`R_3 larr R_3+ R_2`
 `0` `0=-5/2+5/2`
`R_3 larr R_3+ R_2`
 `0` `0=5/2+-5/2`
`R_3 larr R_3+ R_2`


Pivots are the first non-zero element in each row of this eliminated matrix.

`:.` Pivots are `8,5/2`

Here all pivots are positive, so matrix is positive definate.


Test method 2: Determinants of all upper-left sub-matrices are positive.
`A` = 
`8``-6``2`
`-6``7``-4`
`2``-4``3`







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