# Choice of iterative solver for a sparse asymmetric matrix with symmetric structure

I have a sparse $$nxn$$ matrix A with pretty interesting structure. It has a block structure with symmetric structure but asymmetric blocks. Expressed mathematically the block $$A_{jk} = A_{kj}$$ but $$A_{jk} \neq A_{jk}^T$$, so my matrix is not actually symmetric. Are there any algorithms that can take advantage of this kind of system?

First of all - your mathematical description seems wrong: $$A_{jk} = A_{kj}, \forall\ k,j \in \{1,\dots,n\}$$ implies that the matrix is symmetric. So it is not entirely clear what you mean. Also, $$A^T_{jk}$$ is exactly defined via $$A^T_{jk} = A_{kj}$$ (see e.g. Wikipedia). This would somehow be in conflict to your equations.
• @EMP, usually $A_{jk}$ means the $j,k^{\rm th}$ element of matrix $A$ not the block, so some people are confused by your question. There's really no standard notation that I'm aware of to denote a block vs. an element. Aug 17 at 16:32