LU Decomposition

Part 2: More Examples

1. A matrix need not be square to be decomposable into the product of a unit lower triangular matrix L and a matrix U in row echelon form. Enter the 3 x 4 matrix B defined in your worksheet. If B has an LU decomposition, what must the sizes of L and U be? Explain.
2. On a piece of scratch paper, row reduce B by hand to a matrix in row echelon form using only row replacement operations. Use the LU decomposition command to find an LU decomposition of B. Is the announced U the row echelon form of B that you found via row reduction? (If not, use row operations -- as few as possible -- to reduce B to U.)
3. In our general row reduction algorithm we
   • identify a pivot,
   • move it to the correct row,
   • clear the entries beneath the pivot,

   then repeat this process in the next column, and so on. Identify the matrices in your reduction process for B that appear at the pivot identification stages (right before you begin clearing beneath each of the pivots). In each such matrix, circle the newly identified pivot and the entries below that pivot. Column by column, compare your circled entries with the corresponding entries in the unit lower triangular matrix L from your decomposition in Step 2. What do you notice? [Hint: You may want to look at the reduction of A and its corresponding decomposition from Part 1 to verify your observation.]

4. Step 3 gives another method for finding an LU decomposition. Verify this new method on the matrix C defined in your worksheet: Find a row echelon form U of C by row reducing. Find the corresponding unit lower triangular matrix L using your observation from Step 3. Check your answer by comparing C and the product LU.

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