Cholesky Decomposition Example [[6,15,55],[15,55,225],[55,225,979]]

1. Find Cholesky Decomposition ...
[61555155522555225979] $[[6,15,55],[15,55,225],[55,225,979]]$

Solution:

Cholesky decomposition :

A=L⋅LT $A=L*L^T$ , 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 $A$

6 $6$	15 $15$	55 $55$
15 $15$	55 $55$	225 $225$
55 $55$	225 $225$	979 $979$

A matrix is positive definite if Determinants of all upper-left sub-matrices are positive.

Test method 2: Determinants of all upper-left sub-matrices are positive.

A $A$

6 $6$	15 $15$	55 $55$
15 $15$	55 $55$	225 $225$
55 $55$	225 $225$	979 $979$

6 $6$

=6 $=6$

	6 $6$	15 $15$
	15 $15$	55 $55$

=105 $=105$

6 $6$	15 $15$	55 $55$
15 $15$	55 $55$	225 $225$
55 $55$	225 $225$	979 $979$

=3920 $=3920$

Determinants are

6,105,3920 $6,105,3920$

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

Formula

lki=aki-∑i-1j=1lij⋅lkjlii $l_(ki)=(a_(ki) - sum_{j=1}^{i-1} l_(ij) * l_(kj))/(l_(ii))$

lkk=√akk-k-1∑j=1l2kj $l_(kk)=sqrt(a_(kk)-sum_{j=1}^{k-1} l_(kj)^2)$

Here

A $A$

6	15	55
15	55	225
55	225	979

l11=√a11=√6=2.4495 $l_(11)=sqrt(a_(11))=sqrt(6)=2.4495$

l21=a21l11=152.4495=6.1237 $l_(21)=(a_(21))/l_(11)=(15)/(2.4495)=6.1237$

l22=√a22-l221=√55-(6.1237)2=√55-37.5=4.1833 $l_(22)=sqrt(a_(22)-l_(21)^2)=sqrt(55-(6.1237)^2)=sqrt(55-37.5)=4.1833$

l31=a31l11=552.4495=22.4537 $l_(31)=(a_(31))/l_(11)=(55)/(2.4495)=22.4537$

l32=a32-l31×l21l22=225-(22.4537)×(6.1237)4.1833=225-137.54.1833=20.9165 $l_(32)=(a_(32)-l_(31) xx l_(21))/l_(22)=(225-(22.4537)xx(6.1237))/(4.1833)=(225-137.5)/(4.1833)=20.9165$

l33=√a33-l231-l232=√979-(22.4537)2-(20.9165)2=√979-941.6667=6.1101 $l_(33)=sqrt(a_(33)-l_(31)^2-l_(32)^2)=sqrt(979-(22.4537)^2-(20.9165)^2)=sqrt(979-941.6667)=6.1101$

L $L$

l11 $l_(11)$	0 $0$	0 $0$
l21 $l_(21)$	l22 $l_(22)$	0 $0$
l31 $l_(31)$	l32 $l_(32)$	l33 $l_(33)$

2.4495	0	0
6.1237	4.1833	0
22.4537	20.9165	6.1101

L×LT $L xx L^T$

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 $A$

6	15	55
15	55	225
55	225	979

Solution is possible.

This material is intended as a summary. Use your textbook for detail explanation.
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1. Example [61555155522555225979][[6,15,55],[15,55,225],[55,225,979]]

1. Example [61555155522555225979] $[[6,15,55],[15,55,225],[55,225,979]]$