Shel asks:

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Is anyone's symbolic math package able to take the matrix exponential of

[  0  a  0 -d ]
[ -a  0  b  0 ]
[  0 -b  0  c ]
[  d  0 -c  0 ]
>>

I think this can be done pretty easily by hand.  Any matrix M has to satisfy its characteristic equation P(x) = det(xI - M) = 0, where P is a polynomial of deg = 4.

If you write down the first few terms of the series for exp(M), its easy to see how to use P(M) = 0 to express all M^k for k >= 4 in terms of powers of M no greater than the cube.  This will yield exp(M) = aI + bM + cM^2 + dM^3, the only problem being that a,b,c,d will be infinite series.  (For the skew-symmetric matrix above, though, I expect the series will be familiar ones for simple expressions using well-known analytic functions.)  Of course its exp will be a 4D rotation matrix if a,b,c,d are real.

Eigenvalues and eigenvectors should be even easier, since each eigenvector of M with eigenvalue lambda is an eigenvector of exp(M) with eigenvalue exp(lambda). Counting multiplicity I think these constitute all eigenvectors of exp(M).

--Dan