LU Decomposition

If A is an Matrix that can be row reduced to Echelon Form without row exchanges, then

• where is a composition of elementary matrices such that it is a Lower Triangular Matrix where all of its diagonal entries are 1
• is an Echelon Form of

The Process

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

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