 So I want to point out that with this LU factorization, we accomplish it only by doing forward phase replacement operations. What if you can't roll reduce your matrix using those? There are some generalizations to the LU factorization I want to put out for you. So for example, if you do have to do an interchange, whether it's one interchange or 5,208 interchanges or whatever number, right? All of those interchange matrices can be combined together to form a permutation matrix. And so there is a so-called PLU factorization, which the conditions on L and U are exactly the same thing here that the U matrix is a echelon matrix. And if it was square, it would be upper triangular. That's why we call it a U. The matrix L here would be a unit lower triangular matrix that encompasses all of the replacement operations you did in the forward phase. And then P right here is going to be a permutation matrix, which basically we just stick all of the interchanges together in one matrix. And we can get this PLU factorization. So if you have to, whoops, if you have to do a interchange, you can get this PLU factorization, which I want to admit to us that if you slap in a permutation matrix in terms of solving the associated linear system, it's not going to make much of a difference whatsoever. It's not going to affect the factorization by much. So it's a very benign thing to add to it if you have to do interchanges. Now, what if you wanted to do, what if you wanted to do a scaling of some kind, right? An alternative idea is to also have the so-called LDU factorization, which the LDU factorization, as the name might suggest, it has L, which is a unit lower triangular matrix. It has U, which is going to be a echelon matrix, but it also has this matrix D in here, which D is going to be a diagonal matrix, which the diagonal matrix basically keeps all of the scaling together in terms of the row operations we did here because to get the unit lower triangular matrix, we do not need to have any scaling whatsoever. Now, if you did scaling, you can factor them out of the matrix L to make a unit lower triangular. Or in fact, if we wanted the leading entries of the matrix U to be one, we could also factor out factors and make a diagonal matrix. So if we wanted both L and U to have units, have ones in those pivot positions, then we could accomplish that by having a diagonal matrix. So you get this LDU factorization. The last thing to consider here is, there is, you can put all of these things together, and you can get a PLDU factorization. So in the most general sense, every matrix can be factored as a PLDU factorization here. And that's because the permutation matrix P encompasses all of the interchanges. The diagonal matrix D will encompass all of the scaling you have to do. The matrix L is gonna unit lower triangular, it has all the forward replacement matrices there. And then U is this echelon form, which could be upper triangular if it's a square, mind you. And so every funk matrix could have this type of factorization. Now that's a little bit more overkill than what I'm necessarily expecting from this series. So we're gonna focus on the LU factorization, but I did want you to be aware that there are generalizations that allow for these higher type of factorizations to incorporate the diagonal and permutation matrices we talked about previously in this lecture.