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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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