Suppose $r\ll p$, and $A_1,A_2,A_3 \in \mathbf{R}^{rxp}, rank(A_i)=r$. Define minimum singular value as $$\sigma_r(A_i)=\text{r-th singular value of} A_i$$ also define $\kappa_{\min }=\min \left\{\sigma_{r}\left(A_{1}\right), \sigma_{r}\left(A_{2}\right), \sigma_{r}\left(A_{3}\right)\right\}$
My question is following: suppose we combine the matrix $A_i$ by column to form a bigger matrix, i.e. $A=[A_1,A_2,A_3]\in \mathbf{R}^{rx3p}$, why the following hold? $$\sigma_r(A)\asymp \kappa_{\min }$$