Exponential of symmetrical 4x4 matrix

Question

Consider a matrix $A$, such that $A_{ij}=A_{ji}$

I want to find $e^{itA}$

I tried to represent $A$ as sum of 10 different matrices, that would show the symmetry, but the final result ends up to be zero

Am I doing something wrong?

Answer 1

First, diagonalize your matrix $A$ such that $$A=U^{-1}DU$$

Where $U$ is an orthonormal matrix and $D$ is a diagonal matrix. Now use the following identity:

$$\text{Exp}(itU^{-1}DU)=U^{-1}\text{Exp}(itD)U$$

Notice that $\text{Exp}(D)$ is trivial to compute since $D$ is diagonal.