So, another question that came up at tea you guys might like is the following:
Let be matrix with integer entries. When does have integer entries?
One answer is not all that often: Since the eigenvalues of are the exponentials of the eigenvalues of , the eigenvalues of must be algebraic integers (since has integer entries), whose exponentials are algebraic integers. I have no idea whether this has been proved, but common sense suggests the only such algebraic integer is probably 0. Assuming this, we obtain that must be nilpotent. On the other hand, there is a large selection of nilpotent matrices which do have integer exponentials: for example, if all the entries are divisible by where (for example, for all matrices whose square is 0). Can any get these necessary and sufficient conditions a bit closer?