Choosing the inertias for completions of certain partially specified matrices

This paper classifies the ranks and inertias of Hermitian completions of certain band matrices and other partially specified Hermitian matrices with chordal graphs and specified main diagonals. Some results are also presented on positivity- and negativity-preserving Hermitian completions. To complete partially specified Hermitian matrices with chordal graphs the inductive scheme presented by Grone, Johnson, Sa, and Wolkowicz [Linear Algebra Appl., 58 (1984), pp. 109-124] is used. To complete Hermitian band matrices the inductive scheme presented by Dym and Gohberg [Linear Algebra Appl., 36 (1981), pp. 1-24] is used. In both schemes, each inductive step is a one-step completion problem. At each inductive step, the classification of the kernels of one-step completions is used [Linear Algebra Appl., 128 (1990), pp. 117-132]. This allows one to choose both the rank and the inertia of Hermitian completions of certain partial specified Hermitian matrices, while in Johnson and Rodman [Linear and Multilinear Algebra, 16 (1984), pp. 179-195], the rank cannot be chosen and the inertia can be chosen only for maximal rank completions.