LU Decomposition
Part 4: A Modified Decomposition
In this part, we will examine matrices for which our methods for finding an LU decomposition fail. We will modify the LU decomposition algorithm so it works for any matrix.
- Enter the matrix R in your worksheet. As you did in Part 1, multiply R by elementary matrices to reduce it to a matrix U in row echelon form. (Note that a row swap is necessary!) Find a matrix P such that PR = U. Is P unit lower triangular? Explain why or why not.
- Find a matrix L such that R = LU. Have you found an LU decomposition for R? Explain why or why not.
- Ask your computer algebra system to find an LU decomposition of R. Is the product of the reported matrices L and U the same as R? If not, explain how R and the product LU differ. How does this relate to your answers to the previous questions?
- Some texts define the LU decomposition for any matrix by allowing L to be permuted lower triangular, meaning that some permutation of the rows of L can turn L into a unit lower triangular matrix. Determine whether or not your helper application found such a modified LU decomposition for R.
- Enter the matrix S in your worksheet, and use the commands there to reduce S to row echelon form. Use what you have learned to find a modified LU decomposition for S (without using a built-in helper application command). Carefully explain how you found the permuted lower triangular matrix L. Check your answer using the command(s) available in your helper application.
modules at math.duke.edu
