1) Can you clarify what you mean by unit matrix? Unit matrix can have different definitions depending on the problem you're tackling. Do you mean A*inv(A) = I ?
2) The LUDecomposition interface provides access to the pivots in LU decomposition. Other interfaces specific to a decomposition provide their respective pivots. To create an LUDecomposition use the DecompositionFactory_DDRM.
3) No example code for FFT. If you know of a concise enough algorithm I'll add one!
Just updated the page on matrix decomposition. It now mentions how to access information like pivots.