I have seen that often the condition number of a matrix, $A$, may be estimated by taking the QR decomposition of $A$, $A=QR$, and using numerical methods to estimate the condition number of the upper triangular matrix $R$.
When using the definition of the condition number based on the 2-norm, i.e:
$\kappa_2(A)=||A||_2 ||A^{-1}||_2$
This makes sense to me, as:
$||A||_2=||QR||_2=||R||_2$ and $||A^{-1}||_2=||R^{-1}Q'||_2=||R^{-1}||_2$
(by the orthogonality of $Q$ and the fact the 2-norm is unaffected by multiplication by orthogonal matrices).
Therefore, $\kappa_2(A)=\kappa_2(R)$
However, this argument does not seem to hold for $\kappa_1$ or $\kappa_\infty$, the condition numbers based on the 1-norm and infinite-norm, as these norms are affected by multiplication of orthogonal matrices. So my question is:
When estimating $\kappa_1(A)$ or $\kappa_\infty(A)$, what is the logic in calculating $\kappa_1(R)$ or $\kappa_\infty(R)$? Is there some reason why $\kappa_1(A) \approx \kappa_1(R)$ and $\kappa_\infty(A) \approx \kappa_\infty(R)$?
As an example of when this is used, R's kappa and rcond functions estimate condition numbers for $R$ instead of $A$ when calculating the 1-norm and infinite-norm.