Numerical Ranges for Partial Matrices

Abstract

The field of values (or numerical range) is an important notion in the generalization of facts about Hermitian matrices to general complex matrices. Completions of partial Hermitian matrices have now been studied in some depth, and it is time to study completions of more general complex partial matrices. Two natural definitions for the “field of values” of a square partial complex are suggested. The first “builds up” from the inside of any completion, or a specification of the free entries, of a given partial matrix, while the second “whittles down” from the classical field of values of any completion. In general, the resulting sets are different, though there is a universal containment. In this note we characterize those patterns of specified entries for which the two definitions are identical. Chordal graphs again play a key role, but in a somewhat different way from the classical positive definite Hermitian case.

The work of this author was supported in part by National Science Foundation grant DMS-88-02836 and a grant from the National Security Agency.

The work of this author was supported in part by National Science Foundation grant DMS-88-02836 and Office of Naval Research contract N00014-88-K-0661.