A sparse matrix is a matrix that allows special techniques to take advantage of the large number of "background" (commonly zero) elements.

The number of zeros a matrix needs in order to be considered "sparse" depends on the structure of the matrix and the desired operations to perform on it. For example,
a randomly generated sparse matrix with entries scattered randomly throughout the matrix is not sparse
in the sense of Wilkinson (for direct methods) since it takes time to factor (with high probability and for large enough
; Gilbert et al. 1992).

Gilbert, J. R; Moler, C.; and Schreiber, R. "Sparse Matrices in MATLAB: Design and Implementation." SIAM J. Matrix Anal.
Appl.13, 333-356, 1992.Press, W. H.; Flannery, B. P.;
Teukolsky, S. A.; and Vetterling, W. T. "Sparse Linear Systems."
§2.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 63-82, 1992.