Exponentiating integer matrices

So, another question that came up at tea you guys might like is the following:

Let $A$ be matrix with integer entries.  When does $e^A$ have integer entries?

One answer is not all that often: Since the eigenvalues of $e^A$ are the exponentials of the eigenvalues of $A$, the eigenvalues of $A$ must be algebraic integers (since $A$ 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 $A$ 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 $(n-1)!$ where $A^n=0$ (for example, for all matrices whose square is 0).  Can any get these necessary and sufficient conditions a bit closer?

7 thoughts on “Exponentiating integer matrices”

1. ninguem says:

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?

2. David Speyer says:

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.

3. I’ll provide a proof on request.

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

4. 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.

5. Sombra says:

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.