Assume that $\kappa([A, B])$ is the condition number of a block matrix $[A, B]$. Given that, we also know, $$\kappa(C) < \kappa(A)$$
I am curious whether if the following assertion is true or when does that inequality may hold: $$\kappa([C,B]) < \kappa([A, B])$$
Both $A$ and $C$ are $n \times n$ symmetric square matrices and B is an $n\times m$ rectangular matrix.
The condition number of a rectangular matrix $B$ is defined as the ratio of largest and smallest nonzero singular values,
$$\kappa(B) = \frac{\sigma_{\max}(B)}{\sigma_{\min}(B)}$$
where $\sigma_{\max}(B)$ is the largest singular value of $B$ and $\sigma_{\min}(B)$ is the smallest nonzero singular value of matrix $B$.