Suppose you have a QR decomposition of the form:


(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 $(p \times p)$?


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