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9. Cholesky Decomposition example ( Enter your problem )

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 =
 6 15 55 15 55 225 55 225 979

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)
=
 2.4495 0 0 6.1237 4.1833 0 22.4537 20.9165 6.1101

L xx L^T =
 2.4495 0 0 6.1237 4.1833 0 22.4537 20.9165 6.1101
xx
 2.4495 6.1237 22.4537 0 4.1833 20.9165 0 0 6.1101
=
 6 15 55 15 55 225 55 225 979

and A =
 6 15 55 15 55 225 55 225 979

This material is intended as a summary. Use your textbook for detail explanation.
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