A banded matrix
is a square matrix
with zeroes after k elements above and below the main diagonal, where k is less than the size of the matrix (ie if the matrix is nxn then k<n). In the cases where bandedness matter, k is usually significantly
less than n.
Many problems where linear algebra can be applied involves solving large systems of linear equations. In these cases, bandedness, like symmetry, is a good thing. The structure of the matrix can be exploited to significantly lower the time complexity of Gaussian elimination, and save on storage costs.