Inertias of Block Band Matrix Completions

This paper classifies the ranks and inertias of hermitian completion for the partially specified 3 x 3 block band hermitian matrix (also known as a "bordered matrix") P=\pmatrix{A&B&?\cr B^*&C&D\cr ?&D^*&E}. The full set of completion inertias is described in terms of seven linear inequalities involving inertias and ranks of specified submatrices. The minimal completion rank for P is computed. We study the completion inertias of partially specified hermitian block band matrices, using a block generalization of the Dym--Gohberg algorithm. At each inductive step, we use our classification of the possible inertias for hermitian completions of bordered matrices. We show that when all the maximal specified submatrices are invertible, any inertia consistent with Poincare's inequalities is obtainable. These results generalize the nonblock band results of Dancis [SIAM J. Matrix Anal. Appl., 14 (1993), pp. 813--829]. All our results remain valid for real symmetric completions.