LU Decomposition

If A is an $m\times n$ Matrix that can be row reduced to Echelon Form without row exchanges, then

$$A=LU$$

• where $L\in {\mathbb{R}}^{m\times m}$ is a composition of elementary matrices such that it is a Lower Triangular Matrix where all of its diagonal entries are 1
• $U$ is an Echelon Form of $A$

The Process

Suppose A can be row reduced to echelon form U without interchanging rows, i.e.

$$\begin{array}{rlr}{E}_p...E_{0}A& =U& \\ A& =(E_p...E_0{)}^{-1}UA& =LU\end{array}$$

You can construct $L$ by finding each $E$ and multiplying them all together. Alternatively you can construct $L$ such that the sequence of row operations that convert $A$ to $U$ would convert $L$ to $I$