I have been going through the paper by Gu, Ming, and Stanley C. Eisenstat. "Efficient algorithms for computing a strong rank-revealing QR factorization." SIAM Journal on Scientific Computing 17.4 (1996): 848-869 and stumbled on the example concerning the Kahan matrix they give. More precisely, they state:
Let $M = S_n K_n$, where $S_n = \begin{pmatrix} 1 & 0 & \dots & 0 \\ 0 & s & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & s^{n-1} \end{pmatrix}$ and $K_n = \begin{pmatrix} 1 & -c & \dots & -c \\ 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & -c \\ 0 & \cdots & 0 & 1 \end{pmatrix}$ with $c,s > 0$ and $s^2 + c^2 =1$. Let $k=n-1$. Then it can be shown that $$ \frac{\sigma_k(M)}{\sigma_{min}(A_k)} \geq \frac{c^3 (1+c)^{n-4}}{2s}. $$
Here $A_k$ is the leading $k\times k$ principal submatrix of $M$. By $\sigma_i(X)$ we denote the $i$-th largest singular value of the matrix $X$.
I was wondering if someone knows where this estimation is shown in detail and can point me to a reference? Or if someone knows some techniques I could try? My search so far has not been successfull. Thank you!