Find Cholesky Decomposition ...
`[[25,15,-5],[15,18,0],[-5,0,11]]`

Solution:

Here matrix is symmetric positive definite, so Cholesky decomposition is possible.

`A`

=

`25` `15` `-5` `15` `18` `0` `-5` `0` `11`

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`

=

`25` `15` `-5` `15` `18` `0` `-5` `0` `11`

`25`

`=25`

`25`   `15`   `15`   `18`

`=225`

`25`   `15`   `-5`   `15`   `18`   `0`   `-5`   `0`   `11`

`=2025`

Determinants are `25,225,2025`

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

---

---

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)`

Here `A`

=

25 15 -5 15 18 0 -5 0 11

`l_(11)=sqrt(a_(11))=sqrt(25)=5`

`l_(21)=(a_(21))/l_(11)=(15)/(5)=3`

`l_(22)=sqrt(a_(22)-l_(21)^2)=sqrt(18-(3)^2)=sqrt(18-9)=3`

`l_(31)=(a_(31))/l_(11)=(-5)/(5)=-1`

`l_(32)=(a_(32)-l_(31) xx l_(21))/l_(22)=(0-(-1)xx(3))/(3)=(0-(-3))/(3)=1`

`l_(33)=sqrt(a_(33)-l_(31)^2-l_(32)^2)=sqrt(11-(-1)^2-(1)^2)=sqrt(11-2)=3`

So `L`

=

`l_(11)` `0` `0` `l_(21)` `l_(22)` `0` `l_(31)` `l_(32)` `l_(33)`

=

5 0 0 3 3 0 -1 1 3

`L xx L^T`

=

5 0 0 3 3 0 -1 1 3

`xx`

5 3 -1 0 3 1 0 0 3

=

25 15 -5 15 18 0 -5 0 11

and `A`

=

25 15 -5 15 18 0 -5 0 11

Solution is possible.

---

---