The Interplay of Ranks of Submatrices
Gilbert Strang, Tri-Dung Nguyen · SIAM Review · 2004
A banded invertible matrix T has a remarkable inverse. All "upper" and "lower" submatrices of T -1 have low rank (depending on the bandwidth in T). 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 T -1 . Ultimately it reduces to a fact that deserves to be better known: Complementary submatrices of any T and T -1 have the same nullity. The last figure in the paper (when T is tridiagonal) shows that, on and above the diagonal of T -1 , all rows are proportional.