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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+3/4xx8R_2 larr R_2+3/4xx R_1 5/2 5/2=7+3/4xx-6R_2 larr R_2+3/4xx R_1 -5/2 -5/2=-4+3/4xx2R_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=2-1/4xx8R_3 larr R_3-1/4xx R_1 -5/2 -5/2=-4-1/4xx-6R_3 larr R_3-1/4xx R_1 5/2 5/2=3-1/4xx2R_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+0R_3 larr R_3+ R_2 0 0=-5/2+5/2R_3 larr R_3+ R_2 0 0=5/2+-5/2R_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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