I guess I've never run into this pretty simple problem, but I'm doing a Cholesky Factorization of the matrix shown below, and ran into some weird ambiguity in how I usually do $LU$ decomp:
$$ A = \left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 7 \end{array} \right)$$
My first instinct was to do $R_3 = R_3 - R_2$ first, and then $R_2 = R_2 - R_1$ to reduce to $U$, giving me the incorrect L:
$$ L = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{array} \right)$$
I know that we need to start with $R_1$ to get the correct $L$ here, and I know how to do it that way... but why is my first method wrong if I'm row reducing correctly? I've been looking for $LU$ decomp steps but I can't really find any that go beyond saying to "record the steps of your row reductions". Any help would be appreciated.
No, you do not - the selection of row order in which to do the elimination is called "pivoting", and is well-studied.