Suppose I have a matrix $A$ with real entries such that the off-diagonal entries of $A$ are positive or zero. (The diagonal entries may be positive, negative or zero.)
From doing a few examples in Python, it looks like the following might be true of the matrix exponential $e^A$:
The entries of $e^A$ are all real and non-negative (both on and off the diagonal), and
If an entry of $A$ is non-zero, the corresponding entry of $e^A$ will be positive. (For zero entries of $A$, the corresponding entry in $e^A$ might be zero or positive.)
Are these things indeed the case? How can this be shown? Is there a result that will allow me to predict which elements of $e^A$ will be positive, depending on which elements of $A$ are non-zero?