# Sensitivity of the range of a matrix

The distance between two subspaces $\mathcal{U}$ and $\widetilde{\mathcal{U}}$ is classically defined as $d(\mathcal{U},\tilde{\mathcal{U}}):=\|P-\tilde{P}\|$, where $P$ and $\tilde{P}$ are orthogonal projectors on $\mathcal{U}$ and $\tilde{\mathcal{U}}$, and the norm is the Euclidean norm $\|M\|:=\sigma_{\max}(M)$ (largest singular value); see for instance Golub and Van Loan, Matrix Computations, Section 2.6.3.

What is the relative condition number of the image (range, column space) of a matrix $A\in\mathbb{R}^{n\times m}$ with $n>m$ (tall thin matrix) with full column rank, in the subspace metric? I.e., the limit $$k(A)=\lim_{\varepsilon\to 0} \sup_{\|A-\tilde{A}\|\leq \varepsilon\|A\|} \frac{d(\operatorname{im}A,\operatorname{im}\tilde{A})}{\varepsilon}.$$ This should be a close relative of the norm-2 condition number $\kappa(A):=\frac{\sigma_{\max}(A)}{\sigma_{\min}(A)}$. I think I can put together a rough bound by passing through perturbation results for the $Q$ factor of a QR factorization, but I would like to know if this result already appears somewhere in the linear algebra literature.

If the hypotheses of Corollary 3.13 are satisfied (that is, when $\|A_{11}^{-1}\|_2\|E_{11}\|<1$), then we may replace $\hat{\kappa}$ by $\kappa/\gamma$ in (4.1). Thus, $\kappa$ serves as a condition number for $P_A$.
which means that the answer is $\kappa(A)$, with my notation above.