In general, for any two row equivalent matrices A and B, describe how to find a matrix P such that PA = B. Will P always be unit lower triangular? Explain.
What is an LU decomposition of a matrix? If A is an m x n matrix which has an LU decomposition, what are the sizes of L and U?
Under certain conditions, the following assertion is true:
A matrix A can be written as a product A = LU, where U is a row echelon form of A, and L is unit lower triangular.
State the conditions under which this assertion is true, and explain why it is true when the conditions are satisified.
If the conditions you gave in Step 3 are satisfied, explain two ways you can find an LU decomposition for A. [Hint: One method constructs L as a product of elementary matrices, and the other constructs L by inspecting matrices that occur in the process of reducing A].
Assume that A = LU where L is unit lower triangular and U is in echelon form. Explain why solving the two equations
Ly = b and Ux = y
is equivalent to solving Ax = b. Explain why solving the former pair of equations may be more efficient that solving the latter single equation.
If the conditions you gave in Step 3 are not satisfied, we can find a modified LU decomposition for A, where L is permuted unit lower triangular. What is a permuted lower triangular matrix? Explain at least one method for finding a modified LU decomposition for a matrix.
If you ask your helper application to find an LU decomposition for a matrix A, does it always give you a true LU decomposition? If your helper applications sometimes gives a modified LU decomposition, how do you know when an announced decomposition is actually of the modified form?
