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?

e^(non-zero algebraic number) = transcendental was proved by Lindemann in 1882.

http://mathworld.wolfram.com/Hermite-LindemannTheorem.html

Lindemann, 1882: if a is a non-zero algebraic number, then exp(a) is transcendental. Hence, since exp(2i\pi)=1, \pi is transcendental. People don’t learn that these days?

If a matrix with integer entries is nilpotent then there is an integer change of basis which makes it strictly upper triangular (and still have integer entries). I’ll provide a proof on request. So it is enough to answer this question for strictly uper triangular integer matrices. I don’t think there is going to be a very nice answer, though, just a list of congruence conditions.

I, for one, would like to see a proof.

Strike that. I figured it out.

What about the matrix 2^A (or, more properly, exp(A ln 2)?) When does this have integer entries?

If A = [[a b] [b a]], then 2^A = [[2^k + 2^l, 2^k – 2^l] [2^k – 2^l, 2^k + 2^l]] with k = a+b-1, l = a-b-1. (Maple says so.) If a > b this gives an infinite family of solutions.

Thinking about upper triangular matrices, if A = [[a b] [0 d]], then 2^A = [[2^a, b(2^d – 2^a)/(d-a)] [0, 2^d]] and so for any a and d we can find b which makes 2^A have integer entries.

somewhat along the same lines…If a matrix A is nilpotent how do you find A^p for large p? Lambda is 2 with multiplicity 3 and A is upper triangular with 2’s on the diagonal and 1’s above the diagonal.