# Householder matrices for thin QR updating

Suppose you have a QR decomposition of the form:

$$X=QR$$

(Where $$X$$ is an arbitrary matrix of size $$(n \times p)$$, $$Q$$ is an orthogonal matrix of size $$(n \times n)$$ and $$R$$ is an upper trapezoidal matrix of size $$(n \times p)$$).

I have read that you can obtain a QR decomposition of $$X$$ with one of it's rows removed using the original QR decomposition of $$X$$.

Going by this paper, you can use an $$(n \times n)$$ permutation matrix, $$P$$ and an $$(n \times n)$$ Householder matrix $$H$$, which are both dependent on which row is to be removed from $$X$$, in order to calculate:

$$(PQH)(HR) = \begin{bmatrix} 1 & \textbf{0} \\ \textbf{0} & \bar{Q} \end{bmatrix}\begin{bmatrix} v \\ \bar{R} \end{bmatrix}$$

Where $$\bar{Q}$$ and $$\bar{R}$$ are the updated QR decomposition for $$X$$ with the row of interest removed.

However, I was wondering; does anyone know, can this be done for the thin QR decomposition?

I.e. instead of having; $$Q$$ as $$(n \times n)$$ and $$R$$ as $$(n \times p)$$ could this be done for $$Q$$ as $$(n \times p)$$ and $$R$$ as $$(p \times p)$$ using a Householder matrix of dimensions $$(p \times p)$$?