Mathematica does not support this directly. You can do things of this sort using an external package called NCAlgebra.

http://math.ucsd.edu/~ncalg/

The relevant documentation may be found at

http://math.ucsd.edu/~ncalg/DOWNLOAD2010/DOCUMENTATION/html/NCBIGDOCch4.html#x8-510004.4

In particular have a look at "4.4.8 NCLDUDecomposition[aMatrix, Options]"