Determine the solution to a system, given only it's LU decomposition?

Question

Knowing nothing about a particular system of equations, with exception to the system's LU decomposition, how can I interpret the LU decomposed matrices so as to determine the solution to the system?

For example, suppose there exists some unknown system $Ax=B$, someone produces the LU decomposition and gives me $L$ and $U$, how do I "read off" the solutions to, say, $x,y,z$?

Answer 1

If $\mathbf{A}$ admits the LU decomposition $\mathbf{A} = \mathbf{L} \mathbf{U}$, then the linear system of equation $\mathbf{A} \mathbf{x} = \mathbf{b}$ can be written as $\mathbf{b} = \mathbf{L} \mathbf{U} \mathbf{x} = \mathbf{L} \mathbf{y}$ where $\mathbf{y} = \mathbf{U} \mathbf{x}$.

The solution can be then obtained in two steps.

Step 1: Solve for $\mathbf{y}$ using $\mathbf{b} = \mathbf{L} \mathbf{y}$.

Step 2: Solve for $\mathbf{x}$ using the equation $\mathbf{y} = \mathbf{U} \mathbf{x}$.

The main advantage of this method method would be that the system of linear equation in Step 1 & 2 can be easily solved using back-substitution.

Answer 2

$$A=LU$$ becomes $$Ax=(LU)x=L(Ux)$$. Let $$Ux=v$$. Then, we have $Lv=b$. Since you are looking for $x$, you first solve for v by using $Lv=b$. Then using $v$, solve x in the equation $$Ux=v$$. Hope, it helps