The signed enhanced principal rank characteristic sequence

Abstract

The signed enhanced principal rank characteristic sequence (sepr-sequence) of an $n\times n$ Hermitian matrix is the sequence $t_1{t}_2\cdots t_n$, where $t_k$ is either ${\mathtt{A}}^{\ast }$, ${\mathtt{A}}^{+}$, ${\mathtt{A}}^{-}$, $\mathtt{N}$, ${\mathtt{S}}^{\ast }$, ${\mathtt{S}}^{+}$, or ${\mathtt{S}}^{-}$ based on the following criteria: $t_k={\mathtt{A}}^{\ast }$ if $B$ has both a positive and a negative order-$k$ principal minor, and each order-$k$ principal minor is nonzero. $t_k={\mathtt{A}}^{+}$ (respectively, $t_k={\mathtt{A}}^{-}$) if each order-$k$ principal minor is positive (respectively, negative). $t_k=\mathtt{N}$ if each order-$k$ principal minor is zero. $t_k={\mathtt{S}}^{\ast }$ if $B$ has each a positive, a negative, and a zero order-$k$ principal minor. $t_k={\mathtt{S}}^{+}$ (respectively, $t_k={\mathtt{S}}^{-}$) if $B$ has both a zero and a nonzero order-$k$ principal minor, and each nonzero order-$k$ principal minor is positive (respectively, negative). Such sequences provide more information than the $(\mathtt{A},\mathtt{N},\mathtt{S})$ epr-sequence in the literature, where the $k$th term is either $\mathtt{A}$, $\mathtt{N}$, or $\mathtt{S}$ based on whether all, none, or some (but not all) of the order-$k$ principal minors of the matrix are nonzero. Various sepr-sequences are shown to be unattainable by Hermitian matrices. In particular, by applying Muir's law of extensible minors, it is shown that subsequences such as ${\mathtt{A}}^{\ast }\mathtt{N}$ and $\mathtt{N}{\mathtt{A}}^{\ast }$ are prohibited in the sepr-sequence of a Hermitian matrix. For Hermitian matrices of orders $n=1,2,3$, all attainable sepr-sequences are classified. For real symmetric matrices, a complete characterization of the attainable sepr-sequences whose underlying epr-sequence contains $\mathtt{ANA}$ as a non-terminal subsequence is established.

21 pages. The Version of Record of this manuscript has been published, and is available in Linear and Multilinear Algebra (since August 17, 2017) at http://dx.doi.org/10.1080/03081087.2017.1363149

References in corpus (2)

• Classification of families of pr- and epr-sequences
• The enhanced principal rank characteristic sequence over a field of characteristic 2

Cited by in corpus (1)

• The quasi principal rank characteristic sequence