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

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 finitedimensional real vector space with a nondegenerate 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 $M$. 


There is a completely explicit formula in this paper of Bensauod and Mouline (rendicotti Palermo, 2005), which is quite compact for low dimensions. 

