# Solving a linear system with Cholesky factorization

So, I've reached the following problem: Given $A \in R^{nxn}$ positive-definite and symmetrical, $B\in R^{nxn}$ and the vector $c\in R^n$ write an algorithm to solve to following situation:

$Ax=Bc$

After I break down A with Cholesky in $L * L^t$ I don't know how should I proceed, should I get the inverse and find x or is there a easier method?

• It's easy to find the inverse of $L$ because it's already lower-triangular Commented Sep 8, 2017 at 22:20
• It is in the same spirit as using the LU factorization for solving a system. Commented Sep 8, 2017 at 22:29

The idea is the same of LU decomposition, i.e. use the triangular for of the matrix $L$.
For simplicity put, $Bc = b \, \in \mathbb{R}^n$, so the system is: \begin{aligned} Ax &= b \\ L L^{T} x &= b \end{aligned} now you call $L^{T} x = y$ and you solve the system: \left \{ \begin{aligned} Ly &= b \\ L^{T} x &= y \end{aligned} \right.
The matrix $L$ is triangular so you solve it directly with forward and back substitution, first $Ly =b$ and after $L^{T} x = y$.