exponent of a matrix, equivalent conditions

Question

Let $A=[a_{ij}]$ be a real $n\times n$ matrix.

Prove that the following conditions are equivalent:

$(1)$ for every $t\ge 0$, all elements of the matrix $\exp (tA)$ are nonnegative

$(2)$ $a_{ij}\ge 0$ for all $i,j$ with $i\neq j$

Answer 1

$(1)\implies (2)$: According to Andrea , $\lim_{t\rightarrow 0^+}\dfrac{{e^{tA}}_{ij}}{t}=a_{ij}$ and, consequently, $a_{ij}\geq 0$.

$(2)\implies (1)$: There is $s>0$ s.t. $A+sI$ is a non-negative matrix. Then $e^A=e^{-sI}e^{A+sI}=e^{-s}\sum_{n=0}^{\infty}(A+sI)^n/n!$. Thus $e^A$ is a non-negative matrix. In the same way, for every $t\geq 0$, $e^{tA}$ is a non-negative matrix because, for every $i\not= j$, $ta_{ij}\geq 0$.