Question

In the 4th Lecture problems (decomposition into LU), the 2nd problem asks the conditions in which the LU decomposition holds. I'm not sure about some of those - for eg., why is c!=d necessary? They're the last pivot element - which can be 0. Also, if b =a or c =b, all b-a and c-b elements will become zero anyway - so we wouldn't have top worry about doing row exchanges. Am I wrong here?

Answer 1

asanyal, the question asks "Find four conditions on a, b, c, d to get A = LU with four pivots." Although I agree with you that you can do the LU decomposition just fine without imposing conditions, you don't necessarily get 4 pivots (my understanding is that it's not called a pivot if it's zero). Specifically if a=0 then there is a row of zeros; if a=b then rows 1 and 2 are the same; if b=c then rows 2 and 3 are the same; or if c=d then rows 3 and 4 are the same. In any of those cases you're missing at least one pivot out of the four.