Question

3. (a) Compute the Cholesky decomposition of A given by 16 16 -4 -24 16 32-12 -24 -4 -12 301 -24-24 46 by hand (b) Solve the linear system Ax- b, where 16 -63 20 using your Cholesky factorization from Part (a)

Answer 1

.

Here A is a symmetric matrix. For Cholesky decomposition, , where is a lower triangular matrix and $L^T$is its matrix transpose.

So,

Hence,                     

                $f1CwiTwAVFe6dXpBz1UAAAAASUVORK5CYII=$;

$W+GIhwqadLMAAAAASUVORK5CYII=$              

$8tCUAAAAASUVORK5CYII=$                 $Q321519SfP649zi+zqWgAAAABJRU5ErkJggg==$

$fsBL6jOsY+AODLCwyWVQX9AAAAABJRU5ErkJggg=$                      $6UZ9Kv4ACVQr99bi7B8AAAAASUVORK5CYII=$

$ZnQAAAABJRU5ErkJggg==$                     

                      
0 0

                   $78KTUiImp4xbhxqjUxx4AAAAABJRU5ErkJggg==$

$XOAVIm+FpIAAAAASUVORK5CYII=$                     $Qve7re7OvQtMQAAAABJRU5ErkJggg==$

                      

So, which is the Cholesky decomposition.

Now,  

    $HfEAAAAASUVORK5CYII=$

  $n4H49Vvdm+R8tfh+lLjiRGwAAAABJRU5ErkJggg=$

So, we need to solve: along with $Lx=$ .

$VL4ReCiEfgXZjcswuIm6TAAAAAASUVORK5CYII=$

             $jOYmVWH2a2sAAAAASUVORK5CYII=$

Solving we get,

$G8QAAAABJRU5ErkJggg==$

$wGCcfghmCghXgAAAABJRU5ErkJggg==$

Solving we get,

$wBeOUGxm4ROWAAAAABJRU5ErkJggg==$

Add a comment