Solving Linear Systems Using LDL^T Factorization: Step-by-Step Guide

[stunner5000pt]

Use the $LDL^T$ factorization to solve the following linear system
$$\left(\begin{array}{cccc|1}4&1&-1&0&7\\1&3&-1&0&8\\-1&-1&5&2&-4\\0&0&2&4&6\end{array}\right)$$
now i know ihow to get a matrix in the form LDL^T. But i was wondering how one would go about solving from there?



Please help!

 

[HallsofIvy]

Use the $LDL^T$ factorization to solve the following linear system
$$\left(\begin{array}{cccc|1}4&1&-1&0&7\\1&3&-1&0&8\\-1&-1&5&2&-4\\0&0&2&4&6\end{array}\right)$$
now i know ihow to get a matrix in the form LDL^T. But i was wondering how one would go about solving from there?
Please help!

That should be straight forward- the Cholesky decomposition is supposed to be the hard part! L here is a lower triangular matrix, L^T is upper triangular, and D is diagonal, so going from LDL^TX= A to DL^TX= B is just a matter of "back substitution", starting from the value you get immediately in the last row and working up.
Since D is diagonal, going from DL^TX= B to L^TX= C is just dividing by the diagonal elements. Finally, since L^T is upper triangular, going from L^TX= C to X= D is again back substitution, this time working from the top row down.