LU Decomposition: Solving for A

[dirk_mec1]

Homework Statement

http://img296.imageshack.us/img296/6584/93872041bz2.png

Homework Equations

http://img208.imageshack.us/img208/2062/55585340nj2.png

The Attempt at a Solution

I think the key to the solution is the matrix A being non-singular but I can't see how.

 

[cellotim]

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 $$U = L^{-1}L'U'$$. Now you should be able to show that $$I = L^{-1}L'$$ in order to preserve upper-triangularity.

 

[dirk_mec1]

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 $$U = L^{-1}L'U'$$.

Aha, very smart!

Now you should be able to show that $$I = L^{-1}L'$$ in order to preserve upper-triangularity.

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).

If I invert a upper triangular matrix it remains upper triangular so I don't see what you mean by "preserving upper-triangularity".

 

[cellotim]

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.

 

[dirk_mec1]

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.

So to preserve upper triangularity of U w.r.t. U' L^-1 L' should be I?

 

[cellotim]

It needs to be diagonal. It remains to prove that that diagonal matrix must be the identity.