Exponentiating 4 by 4 matrix analytically Does there exist an analytical method by which i can exponentiate a 4 by 4 matrix, in the same way as the general 2 by 2 matrix case in pauli matrix basis. I have dirac matrices (which are composed of direct products of pauli matrices) as my basis for 4 by 4 matrices. 
I need an analytical way !
Any reply is appreciated.
regards
 A: Perhaps I misunderstand the question.  When you say you have Dirac matrices, does that mean that you are computing the exponential of a liner combination of Dirac matrices?  If so, then there is a very simple analytical formula in any dimension: just use the Clifford relations in the exponential series.
More concretely, suppose that you would like to compute the exponential of a matrix $X := \sum_i x^i \Gamma_i$, where the Dirac matrices $\Gamma_i$ obey the Clifford relation
$$
\Gamma_i \Gamma_j + \Gamma_j \Gamma_i = - 2 g_{ij} I~,
$$
with $I$ the identity matrix.  Then it follows from this relation that
$$
X^2 = - x^2 I~,
$$
where I have introduced the (indefinite, if $g_{ij}$ has indefinite signature) "squared norm"
$$
x^2 = \sum_{i,j} x^i x^j g_{ij}~.
$$
If $x^2$ = 0, then
$$
\exp X = I + X
$$
and if $x^2 \neq 0$, then letting $x = \sqrt{x^2}$ (which could be imaginary),
$$
\exp X = \cos x I + \frac{\sin x}{x} X~.
$$
Added (for the "heathens")
Quiaochu's comment is correct.  Here are some more details.
Let $V$ be a finite-dimensional real vector space with a non-degenerate inner product $\left<-,-\right>$.  Let $Cl(V)$ be the corresponding Clifford algebra.  Let $\rho: Cl(V) \to \operatorname{End}(M)$ be an irreducible representation of $Cl(V)$.  Let $(e_i)$ be a basis for $V$.  Then $\Gamma_i := \rho(e_i)$ are called Dirac matrices of $CL(V)$ in the representation $M$.
A: There is a completely explicit formula in this paper of Bensauod and Mouline (rendicotti Palermo, 2005), which is quite compact  for low dimensions.
