Sorry; I should say that it is pretty obvious, but the "obvious proof" is extremely ugly and inelegant. Does anyone have a nicer one?
Let be written in block form
where . Let . Assume that has an decomposition. After steps of Gaussian elimination, we have
.
Prove .
No idea where to start on a non-horrible-looking proof. Has anyone seen this theorem before?
Ok, I have solved the problem. Here is the proof, in case anybody was interested.
Let such that has an decomposition. Perform Gaussian elimination on up to the th step. Then , . By , we have that .
We have that , where
, and
Note that is invertible by triangular with non-zero entries on the main diagonal. Thus we have that
By block multiplication, this implies that
.
But we wish to show that ,
hence we need to show that ,
i.e. ,
where is the inverse of
by .
Thus
.
But
by .
Therefore,
by . Thus the result is proven.