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For a band matrix, elimination without pivoting retains bounded lower and upper bandwidths in its triangular factors. A tridiagonal system therefore needs only linear work and storage, as in the Thomas algorithm. Pivoting can change the precise fill pattern and must be included in the bandwidth and cost analysis when required. The inverse need not be banded, so solving through the factors preserves sparsity that explicit inversion would lose.

A band matrix has nonzero entries confined to a band around its main diagonal. The nonnegative integers $p,q$ in the displayed condition are its lower and upper bandwidths. A tridiagonal matrix has $p=q=1$. Storing only the band reduces memory use; banded Gaussian elimination can exploit the same structure when the pivoting assumptions permit it.