# On a matrix factorization and the Gram-Schmidt process

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 $i<j$, $R_{ij}=\langle e_i, a_j \rangle$. This is spelled out, for example, in the Wikipedia page on QR.

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 $R_{ij} = \langle e_i, a_j \rangle$.

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Think about $AR^{-1}x=Qx$ and $Qx$ is a linear combination of orthogonal vectors. Hence, roughly speaking $R^{-1}$ serves the purpose of transforming the coordinate system into the one which is spanned by the orthogonal columns of $Q$ – user13838 Sep 27 '11 at 1:31