LU Decomposition
Part 1: Basic Decomposition
- Enter the matrix A defined in your worksheet. Reduce A to a matrix U in row echelon form, using only left multiplications by elementary matrices that are lower triangular and have 1's on the main diagonal. (A lower triangular square matrix whose diagonal entries are all 1 is called a unit lower triangular matrix.) There are commands in your worksheet for constructing the first elementary matrix you will need.
- Find a matrix P such that PA = U. If you used only unit lower triangular elementary matrices to reduce A to row echelon form, P will be unit lower triangular. Explain why.
- Explain why a unit lower triangular matrix must be invertible, with an inverse that is also unit lower triangular. [Hint: Think about the row operations you would perform to produce the inverse of a unit lower triangular matrix P by row reducing the matrix [P I].]
- Find a unit lower triangular matrix L such that A = LU. This factorization of A is the LU decomposition of A.
- Check your work in the preceding step by using the LU decomposition command in the worksheet.
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