Define $Re\lambda\_{min}(A)$ to be the minimum of the real parts of the eigenvalues of a matrix $A$. Let $A,B\in \mathbb{R}^{n \times n}$ be two matrice such that $Re\lambda\_{min}(A)>0$, $Re\lambda\_{min}(B)>0$ and $Re\lambda\_{min}(A-B)\geq 0$. Is $Re\lambda\_{min}(A)\geq Re\lambda\_{min}(B)$ right? And how to prove.


Any help will be appreciated!



Inequality regarding the smallest real part of eigenvales
Here is a $2 \times 2$ counterexample:


$A=\begin{bmatrix}10 & 19 \\\\ 8 & 16\end{bmatrix}$


$B=\begin{bmatrix}9 & 2 \\\\ 17 & 7\end{bmatrix}$


If you have tacit assumptions, share them and we'll see if the statement becomes true!


