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

Zhibin Du; Carlos Martins da Fonseca

Czechoslovak Mathematical Journal (2016)

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

Abstract

top
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

top

Du, Zhibin, and da Fonseca, Carlos Martins. "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 Martins},
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 Martins
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

top
  1. Anđelić, M., Erić, A., Fonseca, C. M. da, 10.1080/03081087.2013.794619, Linear Multilinear Algebra 61 (2013), 49-57; erratum ibid. 61 (2013), 1159-1160. (2013) Zbl1315.15006MR3003041DOI10.1080/03081087.2013.794619
  2. Anđelić, M., Fonseca, C. M. da, Mamede, R., 10.1016/j.laa.2010.09.017, Linear Algebra Appl. 434 (2011), 514-525. (2011) Zbl1225.05078MR2741238DOI10.1016/j.laa.2010.09.017
  3. Cvetković, D., Rowlinson, P., Simić, S., 10.1016/0024-3795(93)90491-6, Linear Algebra Appl. 182 (1993), 45-66. (1993) Zbl0778.05057MR1207074DOI10.1016/0024-3795(93)90491-6
  4. Du, Z., The real symmetric matrices with a P-set of maximum size and their associated graphs, J. South China Norm. Univ., Nat. Sci. Ed. 48 (2016), 119-122. (2016) Zbl1363.05159MR3469083
  5. Du, Z., Fonseca, C. M. da, The singular acyclic matrices of even order with a P-set of maximum size, (to appear) in Filomat. 
  6. Du, Z., Fonseca, C. M. da, The acyclic matrices with a P-set of maximum size, Linear Algebra Appl. 468 (2015), 27-37. (2015) Zbl1307.15012MR3293238
  7. Du, Z., Fonseca, C. M. da, 10.1080/03081087.2014.975225, Linear Multilinear Algebra 63 (2015), 2103-2120. (2015) Zbl1334.15026MR3378019DOI10.1080/03081087.2014.975225
  8. Du, Z., Fonseca, C. M. da, Nonsingular acyclic matrices with an extremal number of P-vertices, Linear Algebra Appl. 442 (2014), 2-19. (2014) Zbl1282.15028MR3134347
  9. Du, Z., Fonseca, C. M. da, The singular acyclic matrices with maximal number of P-vertices, Linear Algebra Appl. 438 (2013), 2274-2279. (2013) Zbl1258.05024MR3005289
  10. Erić, A., Fonseca, C. M. da, 10.1016/j.disc.2013.05.018, Discrete Math. 313 (2013), 2192-2194. (2013) Zbl1281.05091MR3084262DOI10.1016/j.disc.2013.05.018
  11. Fernandes, R., Cruz, H. F. da, Sets of Parter vertices which are Parter sets, Linear Algebra Appl. 448 (2014), 37-54. (2014) Zbl1286.15008MR3182972
  12. Horn, R. A., Johnson, C. R., Matrix Analysis, Cambridge University Press, Cambridge (2013). (2013) Zbl1267.15001MR2978290
  13. Johnson, C. R., Duarte, A. Leal, Saiago, C. M., 10.1137/S0895479801393320, SIAM J. Matrix Anal. Appl. 25 (2003), 352-361. (2003) MR2047422DOI10.1137/S0895479801393320
  14. Johnson, C. R., Sutton, B. D., 10.1137/S0895479802413649, SIAM J. Matrix Anal. Appl. 26 (2004), 390-399. (2004) Zbl1083.15015MR2124154DOI10.1137/S0895479802413649
  15. Kim, I.-J., Shader, B. L., 10.1080/03081080701823286, Linear Multilinear Algebra 57 (2009), 399-407. (2009) Zbl1168.15021MR2522851DOI10.1080/03081080701823286
  16. Kim, I.-J., Shader, B. L., 10.1016/j.laa.2007.12.022, Linear Algebra Appl. 428 (2008), 2601-2613. (2008) Zbl1145.15011MR2416575DOI10.1016/j.laa.2007.12.022
  17. Nelson, C., Shader, B., All pairs suffice for a P-set, Linear Algebra Appl. 475 (2015), 114-118. (2015) Zbl1312.15012MR3325221
  18. Nelson, C., Shader, B., Maximal P-sets of matrices whose graph is a tree, Linear Algebra Appl. 485 (2015), 485-502. (2015) Zbl1322.05092MR3394160
  19. Sciriha, I., A characterization of singular graphs, Electron. J. Linear Algebra (electronic only) 16 (2007), 451-462. (2007) Zbl1142.05344MR2365899
  20. Sciriha, I., 10.1016/S0012-365X(97)00036-8, Discrete Math. 181 (1998), 193-211. (1998) Zbl0901.05069MR1600771DOI10.1016/S0012-365X(97)00036-8

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.