Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2308.01397 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929195916984320 |
|---|---|
| author | Martínez-Rivera, Xavier Saejeam, Kamonchanok |
| author_facet | Martínez-Rivera, Xavier Saejeam, Kamonchanok |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_01397 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the signs of the principal minors of Hermitian matrices Martínez-Rivera, Xavier Saejeam, Kamonchanok Rings and Algebras 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. |
| title | On the signs of the principal minors of Hermitian matrices |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2308.01397 |