Diagonally-striped matrices and approximate inverse preconditioners
The inverse of a banded matrix is, in general, dense. If the structure of the original banded matrix is “striped”, that is, the non-zero diagonals are separated by one or more zero diagonals, the inverse may exhibit a similar striped structure. The motivation for studying inverses of striped matrices is to obtain efficient preconditioners for systems arising from radiation transport equations, whose matrices include dominant values along diagonal stripes. Linear systems whose system matrix has a striped inverse lend themselves to the use of a sparse approximate inverse (SPAI) preconditioner whose structure is derived from that of the actual inverse.
Smolarski, D. C. (2006). Diagonally-striped matrices and approximate inverse preconditioners. Journal of Computational and Applied Mathematics, 186(2), 416–431. https://doi.org/10.1016/j.cam.2005.02.012