The real symmetric matrices of odd order with a P-set of maximum size

Zhibin Du; Carlos M. da Fonseca

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 3, page 1007-1026
  • ISSN: 0011-4642

Abstract

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Suppose that A is a real symmetric matrix of order n . Denote by m A ( 0 ) the nullity of A . For a nonempty subset α of { 1 , 2 , ... , n } , let A ( α ) be the principal submatrix of A obtained from A by deleting the rows and columns indexed by α . When m A ( α ) ( 0 ) = m A ( 0 ) + | α | , we call α a P-set of A . It is known that every P-set of A contains at most ⌊ n / 2 ⌋ elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As a first step to the full characterization of these graphs of odd order, we establish some conditions for such graphs G under which there is a real symmetric matrix A whose graph is G and contains a P-set of size ( n - 1 ) / 2 .

How to cite

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Du, Zhibin, and da Fonseca, Carlos M.. "The real symmetric matrices of odd order with a P-set of maximum size." Czechoslovak Mathematical Journal 66.3 (2016): 1007-1026. <http://eudml.org/doc/286817>.

@article{Du2016,
abstract = {Suppose that $A$ is a real symmetric matrix of order $n$. Denote by $m_A(0)$ the nullity of $A$. For a nonempty subset $\alpha $ of $\lbrace 1,2,\ldots ,n\rbrace $, let $A(\alpha )$ be the principal submatrix of $A$ obtained from $A$ by deleting the rows and columns indexed by $\alpha $. When $m_\{A(\alpha )\}(0)=m_\{A\}(0)+|\alpha |$, we call $\alpha $ a P-set of $A$. It is known that every P-set of $A$ contains at most $\lfloor \{n\}/\{2\} \rfloor $ elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As a first step to the full characterization of these graphs of odd order, we establish some conditions for such graphs $G$ under which there is a real symmetric matrix $A$ whose graph is $G$ and contains a P-set of size $\{(n-1)\}/\{2\}$.},
author = {Du, Zhibin, da Fonseca, Carlos M.},
journal = {Czechoslovak Mathematical Journal},
keywords = {real symmetric matrix; graph; multiplicity of eigenvalues; P-set; P-vertices},
language = {eng},
number = {3},
pages = {1007-1026},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The real symmetric matrices of odd order with a P-set of maximum size},
url = {http://eudml.org/doc/286817},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Du, Zhibin
AU - da Fonseca, Carlos M.
TI - The real symmetric matrices of odd order with a P-set of maximum size
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 1007
EP - 1026
AB - Suppose that $A$ is a real symmetric matrix of order $n$. Denote by $m_A(0)$ the nullity of $A$. For a nonempty subset $\alpha $ of $\lbrace 1,2,\ldots ,n\rbrace $, let $A(\alpha )$ be the principal submatrix of $A$ obtained from $A$ by deleting the rows and columns indexed by $\alpha $. When $m_{A(\alpha )}(0)=m_{A}(0)+|\alpha |$, we call $\alpha $ a P-set of $A$. It is known that every P-set of $A$ contains at most $\lfloor {n}/{2} \rfloor $ elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As a first step to the full characterization of these graphs of odd order, we establish some conditions for such graphs $G$ under which there is a real symmetric matrix $A$ whose graph is $G$ and contains a P-set of size ${(n-1)}/{2}$.
LA - eng
KW - real symmetric matrix; graph; multiplicity of eigenvalues; P-set; P-vertices
UR - http://eudml.org/doc/286817
ER -

References

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