Product of atomic triangular matrices
I am trying to prove the following: I am having a triangular matrix (with ), I can create atomic triangular matrices (in other words: For I set all elements that are not on the main diagonal and that are not in column to be zero.) Then I need to show that this is true:
So far I have found: (what might even be the wrong attempt)
where: is an Element of the result after I performed the first multiplications. and is an Element of the atomoc triangular matrix .
However, this is where I am stuck: I cannot figure out a way how to prove by the last equation that all the elements must be as in the original matrix.
Thank you for ideas / suggestions / solutions!