Some properties of the matrix exponential

We give a simple condition on a matrix A for which if the exponential matrix e^A is diagonal, lower or upper triangular, then so is A. It is also shown that for diagonalizable A and any matrix B, e^A and B commute if and only if A and B commute. These results are useful in problems in which knowledge about A has to be extracted from structural information about its exponential, such as in large-scale sampled-data systems. They also find application in the area of blind system identification