Orthogonality and QR factorization

Question

I need help with part B

I'm guessing since $A^TA = LL^T$, and $R^TR = LL^T$, then $R=L^T$ or $R=L$ but since it can't be defined which one, it is false? I'm I correct? who would I elaborate on this?

Answer 1

A first reason for which one cannot conclude $R=L^T$ is that it is always possible to insert a diagonal matrix $D$ with diagonal entries $\pm1$ and its inverse $D^{-1}=D^T$ in such a way that:

$$R^TD^TDR=LL^T$$

This identity can be written :

$$(DR)^TDR=LL^T$$

with $DR$ being still an upper diagonal matrix ; thus, assuming that we could identify $R=L^T$, we could as well make identification $DR=L^T$ which is incompatible with $R=L^T$...