Given a real square matrix $A$, we can factor it as $A = QR$ where $Q$ is orthogonal and $R$ is upper triangular. The entries of $R$ have a simple geometric interpretation in terms of the vectors one gets doing the Gram-Schmidt process on the columns of $A$. In particular, if $a_i$ is the $i$'th column of $A$ and $e_j$ is the $j$'th vector produced by the Gram-Schmidt process, then for $i
My question: Suppose $A$ is full rank and so $R$ is invertible. What interpretation, if any, do the entries of $R^{-1}$ have?
My motivation: I need to work out something about $R^{-1}$ if the columns of $A$ satisfy a certain property. Its a bit involved to go into here, but any way I could reason about the entries of $R^{-1}$ in terms of the geometry of the columns of $A$ would be helpful.
Edited: Perhaps I should say that I do realize that $AR^{-1}=Q$. In other words, the Gram-Schmidt process produces linear combinations of the columns of $A$ that are orthogonal, and the coefficients of those linear combinations are precisely in the columns of $R^{-1}$. However, I'm still wondering if a more direct geometric interpretation can be given - something like $R_{ij} = \langle e_i, a_j \rangle$.