A matrix is sparse when most of its entries are zero. Large discretized physical models often produce sparse systems because each unknown interacts directly with only a small number of nearby unknowns.

Storing every matrix entry wastes memory and computation. Sparse representations store only nonzero values together with enough information to locate them.

The sparsity pattern records which entries can be nonzero. This pattern often reflects the connectivity of the underlying mesh, graph or model.

Sparse matrix-vector products can exploit this structure directly. Direct elimination can create new nonzero entries, called fill-in, so the ordering of unknowns influences memory use and cost.

Iterative solvers are also attractive for many large sparse systems because they can work primarily through matrix-vector operations without forming dense intermediate matrices.

Sparsity is therefore a computational property of a linear system that can be as important as its mathematical dimension.