The Enhanced Principal Rank Characteristic Sequence for Hermitian Matrices

Abstract

The enhanced principal rank characteristic sequence (epr-sequence) of an $n\x n$ matrix is a sequence $\ell_1 \ell_2 \cdots \ell_n$, where each $\ell_k$ is ${\tt A}$, ${\tt S}$, or ${\tt N}$ 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 ${\tt N}$s with a gap in between. Hermitian adjacency matrices of mixed graphs that begin with ${\tt NAN}$ are characterized. All attainable epr-sequences of Hermitian matrices of orders $2$, $3$, $4$, and $5$, are listed with justifications.