A question from my textbook:
"Which invertible matrices allow A = LU (elimination without row exchanges)? Good question! Look at each of the upper left submatrices of A.
All upper left k x k submatrices A_{k} must be invertible (sizes k = 1,...,n).
Explain that answer: A_{k} factors into _____ because LU = {{L_{k},0},{*,*}}{{U_{k},0},{0,*}}"
I really not sure what this is saying. I can sort of understand how elimination can be seen as a factorization in which A = LU, U being upper triangular and L being the elimination matrix, but I don't see how it applies here. I'm not really sure what the equation is trying to get me to think about and I'm not sure why the statement about all upper left matrices is right? Is it because there are all pivots?