The Enhanced Principal Rank Characteristic Sequence for Hermitian Matrices

The enhanced principal rank characteristic sequence (epr-sequence) of an n×n matrix is a sequence `1`2 · · ·`n, where each `k is A, S, or N according as all, some, or none of its principal minors of order k are nonzero. There has been substantial work on epr-sequences of symmetric matrices (especially real symmetric matrices) and real skew-symmetric matrices, and incidental remarks have been made about results extending (or not extending) to (complex) Hermitian matrices. A systematic study of epr-sequences of Hermitian matrices is undertaken; the differences with the case of symmetric matrices are quite striking. Various results are established regarding the attainability by Hermitian matrices of epr-sequences that contain two Ns with a gap in between. Hermitian adjacency matrices of mixed graphs that begin with NAN are characterized. All attainable epr-sequences of Hermitian matrices of orders 2, 3, 4, and 5, are listed with justifications.