ReduceRow of one matrix

Questions
1

Salem,
I want to reduce the row of one matrix (attachment file).
I want to use the Gauss method to will have one matrix like this :
first line * * * *;
second line 0 * * *;
third line 0 0 * *;
fourth line 0 0 0 *;
I use the ReduceRow option, but the result is not good.
The example is sent as an attachment of this post.
I am waiting for your help. Greetings.

2

may be what you want is LUDecomposition

{lu, p, c} = LUDecomposition[H];
u = lu SparseArray[{i_, j_} /; j >= i -> 1, {4, 4}] // MatrixForm
3

thank you for this response, I think that this is what I want to use, I will check it.