For a symmetric state space system $G(s)=\left\{A,B,C,D\right\}$, the cross Gramain matrix $R$ is the solution of $$AR+RA+BC=0$$
Using eigenvalue decomposition, problem is to obtain a matrix U which diagonalizes the cross gramian matrix $R$, resulting a diagonal matrix $S$ such that $$S=U^{-1}RU$$
Note: For state space symmetric system, $A=A^T, C=B^T$ where $T$ is the transpose of a matrix.
Could you specify your question on the cross Gramian?
You can find more information about the meaning of the eigen-decomposition of the cross Gramian matrix in the article "Cross-Gramian Based Model Reduction for Data-Sparse Systems" ( http://emis.ams.org/journals/ETNA/vol.31.2008/pp256-270.dir/pp256-270.pdf ) and references therein.