Mathematics Asked by LudvigH on October 19, 2020

I can find sources on the internet that provides expression of the derivative of $Q$ and $R$ with respect to $A$ in the expression $A=QR$.

One such example is https://j-towns.github.io/papers/qr-derivative.pdf. However, I have not yet found a proof on whether $QR$ factorization of $A$ is continous in $A$. I guess this must be the case, since people take the time to diffrentiate it, and that wikipedia claim that it is numerically stable.

Let $(A_k)_{k=1}^{infty}$ be a series of real $mtimes n$ matrices. By QR decomposition, define $Q_k$ and $R_k$ by $A_k=Q_kR_k$. Let $lim_{kto infty}A_k = A$ for some real $mtimes n$ matrix $A$. Let $A=QR$ by QR factorization.

Is it so that $lim_{kto infty}Q_k = Q$ and $lim_{kto infty}R_k = R$? Why? Does it matter whether all the matrices $A_k$ and $A$ has the same rank or not?

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