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.


1 Answer 1


Quick self-answer for future reference: there is a brief statement of this result on Stewart, Sun, Matrix perturbation theory, p. 154:

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.