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Problem: cholesky decomposition [[6,15,55],[15,55,225],[55,225,979]] [ Calculator, Method and examples ]

Solution:
Your problem `->` cholesky decomposition [[6,15,55],[15,55,225],[55,225,979]]


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 definite, so Cholesky decomposition is possible.

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

`A` = 
`6``15``55`
`15``55``225`
`55``225``979`


Test method 1: Existence of all Positive Pivots.
First apply Gaussian Elimination method to find Pivots
`A` = 
`6``15``55`
`15``55``225`
`55``225``979`


`R_2 larr R_2-5/2xx R_1`

 = 
`6``15``55`
`0``35/2``175/2`
`55``225``979`


`R_3 larr R_3-55/6xx R_1`

 = 
`6``15``55`
`0``35/2``175/2`
`0``175/2``2849/6`


`R_3 larr R_3-5xx R_2`

 = 
`6``15``55`
`0``35/2``175/2`
`0``0``112/3`


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

`:.` Pivots are `6,35/2,112/3`

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


Test method 2: Determinants of all upper-left sub-matrices are positive.
`A` = 
`6``15``55`
`15``55``225`
`55``225``979`


 `6` 
`=6`


 `6`  `15` 
 `15`  `55` 
`=105`


 `6`  `15`  `55` 
 `15`  `55`  `225` 
 `55`  `225`  `979` 
`=3920`


Determinants are `6,105,3920`

Here all determinants are positive, so matrix is positive definite.


A matrix is positive definite if it's symmetric and all its eigenvalues are positive.

Test method 3: All positive eigen values.
`|A-lamdaI|=0`

 `(6-lamda)`  `15`  `55` 
 `15`  `(55-lamda)`  `225` 
 `55`  `225`  `(979-lamda)` 
 = 0


`:.(6-lamda)((55-lamda) × (979-lamda) - 225 × 225)-15(15 × (979-lamda) - 225 × 55)+55(15 × 225 - (55-lamda) × 55)=0`

`:.(6-lamda)((53845-1034lamda+lamda^2)-50625)-15((14685-15lamda)-12375)+55(3375-(3025-55lamda))=0`

`:.(6-lamda)(3220-1034lamda+lamda^2)-15(2310-15lamda)+55(350+55lamda)=0`

`:. (19320-9424lamda+1040lamda^2-lamda^3)-(34650-225lamda)+(19250+3025lamda)=0`

`:.(-lamda^3+1040lamda^2-6174lamda+3920)=0`

`:.-(lamda^3-1040lamda^2+6174lamda-3920)=0`

`:.(lamda^3-1040lamda^2+6174lamda-3920)=0 `

Roots can be found using newton raphson method
Newton Raphson method for `x^3-1040x^2+6174x-3920=0`


Here `x^3-1040x^2+6174x-3920=0`

Let `f(x) = x^3-1040x^2+6174x-3920`

`:. f'(x) = 3x^2-2080x+6174`

`x_0 = 0`


`1^(st)` iteration :

`f(x_0)=f(0)=1 xx 0^3-1040 xx 0^2+6174 xx 0-3920=-3920`

`f'(x_0)=f'(0)=3 xx 0^2-2080 xx 0+6174=6174`

`x_1 = x_0 - f(x_0)/(f'(x_0))`






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