The Interplay of Ranks of Submatrices

A banded invertible matrix has a remarkable inverse. All “upper” and “lower” submatrices of have low rank (depending on the bandwidth in ). The exact rank condition is known, and it allows fast multiplication by full matrices that arise in the boundary element method. We look for the “right” proof of this property of . Ultimately it reduces to a fact that deserves to be better known: Complementary submatrices of any and have the same nullity. The last figure in the paper (when is tridiagonal) shows two submatrices with the same nullity . Then has rank . On and above the diagonal of , all rows are proportional.