Let us define the complex matrix $X \in \mathbb{C}^{2N \times M}$ where $N > M$. Additionally, the matrix ${X}$ consists of the following submatrices: $$ {X} = \left[ \begin{array}[c]. A \\ B\end{array} \right] $$ where $A \in \mathbb{C}^{N \times M}$ and $B \in \mathbb{C}^{N \times M}$ and their columns are linearly independent.
Is there a way I could show that if $A=B$, the condition number of $X$ would be larger than the condition number of $X'$ with $A \neq B$ (particularly if $A$ and $B$ are linearly independent)?
Here is an answer for the condition number relative to the $2$-norm. We have $$ \sigma_{\max}(X) \leq \sqrt{\sigma_{\max}^2(A) + \sigma_{\max}^2(B)}, \quad \sigma_{\min}(X) \geq \sqrt{\sigma_{\min}^2(A) + \sigma_{\min}^2(B)}. $$ From this, it follows that $$ \kappa(X) = \frac{\sigma_\max(X)}{\sigma_\min(X)} \leq \sqrt{\frac{\sigma_{\max}^2(A) + \sigma_{\max}^2(B)}{\sigma_{\min}^2(A) + \sigma_{\min}^2(B)}}. $$ Moreover, this inequality will be equality in the case that $A$ and $B$ are linearly dependent.