I'm doing oblique projections of matrices $A \in \Re^{pxj}, B \in \Re^{qxj}, C \in \Re^{rxj}$. The formula look like this:
$$A/_{B}C = A(C^T \space\space\space\space B^T).[(\frac{CC^T}{BC^T} \space\space\space \frac{CB^T}{BB^T})^{\dagger}]_{first\space\space r \space\space columns} .C$$
Correct me if I'm wrong. But as I know, the formula contains element wise multiplication. The formula is from "Subspace identification for linear systems, page 21, Peter van Overschee". Not sure if this is right because other documents does not use element wise multiplication.
Like this one: http://www.trnka.name/download/subspace/ThesisProposal.pdf (page 6)
Question: So...I know $r$ but I cannot interpret the sentence "first r columns". Is it the only one column at index $r$ or is it all columns from 1 to column $r$?
Matlab notation:
>> B = B(:, 1:r), C = C(:, 1:r)
If it would have been the column at index $r$ then the common way of phrasing it would be "the $r$th column". But the "first $r$ columns" refers to the columns (so plural) from index 1 to $r$.
Another way of writing same equation, without this wording, would be
$$ \underset{B}{A/C} = A \begin{bmatrix} C^\top & B^\top \end{bmatrix} \left( \begin{bmatrix} C \\ B \end{bmatrix} \begin{bmatrix} C^\top & B^\top \end{bmatrix} \right)^\dagger \begin{bmatrix} C \\ O \end{bmatrix} $$
with $O \in \mathbb{R}^{q \times j}$ a zero matrix (so $O = 0\,B$).