Minimal rank completions of partial banded matrices

It is proven that the minimal rank of a partial banded matrix equals the maximum of the minimal ranks of all triangular subpatterns. This proves partly the minimal rank conjecture in a paper by N. Cohen, C. R. Johnson, L. Rodman and H. J. Woerdeman (Operator Theory: Advances and Applications 40 (1989), 165-185).

The results are applied to the problem of simultaneously completing a matrix and its inverse.