After QR decomposition of a matrix, $M$, the columns of Q are orthonormal. Is it possible after obtaining *Q*, we recover unnormalized column vectors from $Q$? For example, the matrix M has the following $Q$ matrix.

$M= \begin{bmatrix}1&-1&4\\1&4&-2\\1&4&2\\1&-1&0 \end{bmatrix}$

The $Q$ matrix is

$Q=\begin{bmatrix}0.5& -0.5& 0.5\\0.5& 0.5& -0.5\\0.5& 0.5& 0.5\\0.5&-0.5&-0.5 \end{bmatrix}$

This is for a certain application in spectroscopy. The process is as follows using Gram-Schmidt method.

The raw data is an experimental spectroscopy data. In general, we will have background signals, call them v1, v2, v3 as column vectors. Now measure the spectrum of the molecule of interest. This vector is m. Orthonormalize v1,v2 and v3 to obtain e1, e2 and e3. The magnitude of e1,e2,and e3 is 1. I am following the notation of Wikipedia's article on Gram-Schmidt Wiki.

Then here comes the key step: After an orthonormal basis set of e1, e2, and e3 has been obtained, from the background spectra, measure the spectrum of molecule of interest. This vector is $m$.

Now orthogonalize $m$, with respect to e1,e2, and e3 to obtain another vector $u$. In other words, the dot product u.e1=u.e2=u.e3 = 0.

Since $u$ has not been normalized, i.e., it has a length given by sqrt($u.u$). The length of $u$ is of interest for measurement purposes.

I wanted to know if this is possible by QR method or not?