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 matri ces with chordal graphs and specified main diagonals. Some results are also presented on positivity- and negativity-preserving Hermitian com pletions. To complete partially specified Hermitian matrices with chor dal graphs the inductive scheme presented by Grone, Johnson, Sa, and W olkowicz [Linear Algebra Appl., 58 (1984), pp. 109-124] is used. To co mplete Hermitian band matrices the inductive scheme presented by Dym a nd Gohberg [Linear Algebra Appl., 36 (1981), pp. 1-241 is used. In bot h schemes, each inductive step is a one-step completion problem. At ea ch inductive step, the classification of the kernels of one-step compl etions is used [Linear Algebra Appl., 128 (1990), pp. 117-132). This a llows one to choose both the rank and the inertia of Hermitian complet ions 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 maxim al rank completions.