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.
LU decomposition is a method used to solve linear equations by breaking down a matrix into a lower triangular matrix and an upper triangular matrix. It is useful because it simplifies the process of solving large systems of equations, making it more efficient and easier to understand.
LU decomposition is different from other methods of solving linear equations, such as Gaussian elimination or Cramer's rule, because it breaks down a matrix into two triangular matrices, making it easier to solve and more efficient for larger systems of equations.
No, LU decomposition is not always possible for any given matrix. The matrix must be non-singular, meaning it has a non-zero determinant, and it must not have any zero pivots in the Gaussian elimination process.
The process for using LU decomposition to solve for A involves first decomposing the original matrix into a lower triangular matrix and an upper triangular matrix. Then, using back substitution, you can solve for the variables in the lower triangular matrix and then use those values to solve for the variables in the upper triangular matrix.
The advantages of using LU decomposition over other methods of solving linear equations include its efficiency for larger systems of equations, its ability to handle non-square matrices, and the fact that it can be used for repeated solutions with different right-hand sides without needing to repeat the decomposition process.