Let $A$ e $B$ be two symmetric matrices, with the additional property that the sum of all the entries of any column is zero.
If the $k-$th column of $A$ equals the $k-$th column of $B$, what can be said about the $k-$th columns of $\exp(A)$ and $\exp(B)$?
From some tinkering in Mathematica, it looks like the equality should be preserved even after the exponentiation. If that is indeed the case, how does one go about proving it?
This is false. For instance, let $$A=\begin{pmatrix} 1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 0\end{pmatrix}$$ and $$B=\begin{pmatrix} 1 & -1 & 0 \\ -1 & 0 & 1 \\ 0 & 1 & -1\end{pmatrix}.$$ These have the same first column but $\exp(A)$ and $\exp(B)$ do not have the same first column.