The discussion revolves around LU decomposition of a matrix A, specifically addressing the implications of A being non-singular and the properties of the resulting matrices L and U. Participants are exploring the relationships between different LU decompositions and the conditions necessary for maintaining upper-triangularity.
The discussion is active, with participants providing insights and questioning assumptions about the definitions and properties of the matrices involved. Some guidance has been offered regarding the preservation of upper-triangularity, but there is no explicit consensus on the implications of the M-matrices or the necessity of the identity matrix in the context of the discussion.
Participants are considering the definitions of L and U in relation to the M-matrices, and there is an ongoing examination of the conditions under which the diagonal matrix must be the identity. The discussion is framed within the constraints of homework rules, focusing on understanding rather than providing direct solutions.
Aha, very smart!cellotim said:Suppose we have two different LU decompositions, A = LU and A=L'U'. Because A is non-singular L, U, L' and U' are all non-singular and invertible. This implies that [tex]U = L^{-1}L'U'[/tex].
Well...from my instructors note the L is defined in a product of M-matrices(see picture) but if that's the case then L and L' have different M's (for we presume L and L' to be different).Now you should be able to show that [tex]I = L^{-1}L'[/tex] in order to preserve upper-triangularity.
cellotim said:The M's are not necessary as far as I can see only that L and L' are lower-triangular and U and U' are upper-triangular, and I mean preserve upper-triangularity of U' to U.