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 $t_1t_2 \cdots t_n$, where $t_k$ is $\tt A^*$, $\tt A^+$, $\tt A^-$, $\tt N$, $\tt S^*$, $\tt S^+$, or $\tt S^-$, based on the following criteria: $t_k = \tt A^*$ if all the order-$k$ principal minors of $B$ are nonzero, and two of those minors are of opposite sign; $t_k = \tt A^+$ (respectively, $t_k = \tt A^-$) if all the order-$k$ principal minors of $B$ are positive (respectively, negative); $t_k = \tt N$ if all the order-$k$ principal minors of $B$ are zero; $t_k = \tt S^*$ if $B$ has a positive, a negative, and a zero order-$k$ principal minor; $t_k = \tt S^+$ (respectively, $t_k = \tt S^-$) 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.