Sparse matrix

In numerical analysis, a sparse matrix is a matrix in which most of the elements are zero. By contrast, if most of the elements are nonzero, then the matrix is considered dense. The fraction of non-zero elements over the total number of elements (i.e., that can fit into the matrix, say a matrix of dimension of m x n can accommodate m x n total number of elements) in a matrix is called the sparsity (density).

Conceptually, sparsity corresponds to systems which are loosely coupled. Consider a line of balls connected by springs from one to the next: this is a sparse system as only adjacent balls are coupled. By contrast, if the same line of balls had springs connecting each ball to all other balls, the system would correspond to a dense matrix. The concept of sparsity is useful in combinatorics and application areas such as network theory, which have a low density of significant data or connections.

Large sparse matrices often appear in scientific or engineering applications when solving partial differential equations.