On the signs of the principal minors of Hermitian matrices

Abstract

The signed enhanced principal rank characteristic sequence (sepr-sequence) of a given $n\times n$ Hermitian matrix B is the sequence $t_1{t}_2\dots t_n$, where $t_k$ is ${\mathtt{A}}^{\mathtt{\ast }}$, ${\mathtt{A}}^{\mathtt{+}}$, ${\mathtt{A}}^{\mathtt{-}}$, $\mathtt{N}$, ${\mathtt{S}}^{\mathtt{\ast }}$, ${\mathtt{S}}^{\mathtt{+}}$, or ${\mathtt{S}}^{\mathtt{-}}$, based on the following criteria: $t_k={\mathtt{A}}^{\mathtt{\ast }}$ if all the order-k principal minors of B are nonzero, and two of those minors are of opposite sign; $t_k={\mathtt{A}}^{\mathtt{+}}$ (respectively, $t_k={\mathtt{A}}^{\mathtt{-}}$) if all the order-k principal minors of B are positive (respectively, negative); $t_k=\mathtt{N}$ if all the order-k principal minors of B are zero; $t_k={\mathtt{S}}^{\mathtt{\ast }}$ if B has a positive, a negative, and a zero order-k principal minor; $t_k={\mathtt{S}}^{\mathtt{+}}$ (respectively, $t_k={\mathtt{S}}^{\mathtt{-}}$) if B has both a zero and a nonzero order-k principal minor, and all the nonzero order-k principal minors of B are positive (respectively, negative). A complete characterization of the sequences of order 2 and order 3 that do not occur as a subsequence of the sepr-sequence of any Hermitian matrix is presented (a sequence has order k if it has k terms). An analogous characterization for real symmetric matrices is presented as well.

Keywords:

• Principal minor
• Hermitian matrix
• real symmetric matrix
• signed enhanced principal rank characteristic sequence
• enhanced principal rank characteristic sequence

AMS CLASSIFICATIONS:

• 15A15
• 15B57
• 15A03

Acknowledgments

Kamonchanok Saejeam's research was supported, in part, by a Summer Research Apprenticeship grant from the Dean of the Faculty's Office at Bates College.

Disclosure statement

No potential conflict of interest was reported by the author(s).