# Fill-in

### From Glossary

Arises in the context of sparse matrices
subjected to certain operations. In particular, a basis may be
factored in a product of elementary matrices
to represent Gaussian elimination. The nonzeroes
that appear in positions that were not in the original matrix are called *fill-in*
coefficients. An example of a matrix that has no fill-in for this factorization
is one that is lower triangular. In that case the
factors appear as:

On the other hand, a lower triangular matrix could cause fill-in for some other factorization, such as the Cholesky factorization. A completely dense matrix also has no fill-in since there were no zeroes to begin with. Here is an example of fill-in, taking the original order of rows and columns in the product form:

where is fill-in.