Wolfram Alpha:

```Matrix Exponential
------------------

The matrix exponential is an exponential function where
the exponential of M is the SQUARE n x n real or complex
matrix.  It is given by the power series:

exp(M) = 1 + M + M2/2! ...

∞
= ΣMk/k!
k=0

Consider the matrix for a rotation about the x axis:

-            -
R = | 1   0     0  |
| 0 cosθ -sinθ |
| 0 sinθ  cosθ |
-            -

-                                 -
= | 1       0                0         |
| 0 (1 - θ2/2! ...) -(θ - θ3/3! ...) |
| 0 (1 - θ3/3! ...)  (1 - θ2/2! ...) |
-                                 -

For small θ we get:

-      -
R = | 1 0  0 |
| 0 1 -θ |
| 0 θ  1 |
-      -

-      -
= I + | 0 0  0 | + ...
| 0 0 -θ |
| 0 θ  0 |
-      -

-      -
= I + θ| 0 0  0 | + ...
| 0 0 -1 |
| 0 1  0 |
-      -

Determinant of the Matrix Exponential
-------------------------------------

The determinant of the ME is given by the JACOBI FORMULA.

det(exp(A)) = exp(Tr(A))

Matrix Exponential of Sums
--------------------------

If the matrices A and B commute (meaning that AB = BA),
then:

exp(A)exp(B) = exp(A + B) = 1 + (A + B) + (A + B)2/2! ...

If they don't commute then we use the BAKER-CAMPBELL
-HAUSEDORFF formula:

exp(A)exp(B) = exp(A + B + [A,B]/2 + ([A,[A,B]] + [B,[B,A]])/12 ...)

Where there is no known closed form expression for
this result.

Diagonal Matrix
---------------

If G is a diagonal matrix then:

exp(G) = 1 + G + G2/2 + ...

-       -     -           -           -               -
= | 1 0 0 0 |   | g1 0  0  0  |         | g12   0   0   0  |
| 0 1 0 0 | + | 0  g2 0  0  | + (1/2!)|  0  g22   0   0  | + ...
| 0 0 1 0 |   | 0  0  g3 0  |         |  0   0  g32   0  |
| 0 0 0 1 |   | 0  0  0  g4 |         |  0   0   0   g42 |
-       -     -           -           -               -

-                               -
= | exp(g1)   0       0       0     |
|   0     exp(g2)   0       0     |
|   0       0     exp(g3)   0     |
|   0       0       0     exp(g4) |
-                               -
```