Skip to main content
Erschienen in: Journal of Applied Mathematics and Computing 1-2/2013

01.07.2013 | Applied mathematics

Ranks of submatrices in (skew-)Hermitian solutions to a quaternion matrix equation

verfasst von: Ning Li, Jing Jiang, Guang-Jing Song

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2013

Einloggen

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Assume that X and Y are Hermitian solutions to quaternion matrix equation AXA +BYB =C, which are partitioned into 2×2 block forms. We in this paper give the maximal and minimal ranks of submatrices in the Hermitian solutions X=X , Y=Y , and establish necessary and sufficient conditions for the submatrices to be zero, unique as well as independent. The findings of this paper widely extend the known results in the literature.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Baksalary, J.K.: Nonnegative definite and positive definite solutions to the matrix equation AXA ∗=B. Linear Multilinear Algebra 16, 133–139 (1984) MathSciNetMATHCrossRef Baksalary, J.K.: Nonnegative definite and positive definite solutions to the matrix equation AXA =B. Linear Multilinear Algebra 16, 133–139 (1984) MathSciNetMATHCrossRef
3.
Zurück zum Zitat Chang, X.W., Wang, J.S.: The symmetric solution of the matrix equations AX+YA=C, AXA T +BYB T =C, and (A T XA,B T XB)=(C,D). Linear Algebra Appl. 179, 171–189 (1993) MathSciNetMATHCrossRef Chang, X.W., Wang, J.S.: The symmetric solution of the matrix equations AX+YA=C, AXA T +BYB T =C, and (A T XA,B T XB)=(C,D). Linear Algebra Appl. 179, 171–189 (1993) MathSciNetMATHCrossRef
5.
Zurück zum Zitat Dai, H., Lancaster, P.: Linear matrix equations from an inverse problem of vibration theory. Linear Algebra Appl. 246, 31–47 (1996) MathSciNetMATHCrossRef Dai, H., Lancaster, P.: Linear matrix equations from an inverse problem of vibration theory. Linear Algebra Appl. 246, 31–47 (1996) MathSciNetMATHCrossRef
6.
Zurück zum Zitat Groß, J.: A note on the general Hermitian solution to AXA ∗=B. Bull. Malays. Math. Soc. 21, 57–62 (1998) MATH Groß, J.: A note on the general Hermitian solution to AXA =B. Bull. Malays. Math. Soc. 21, 57–62 (1998) MATH
7.
Zurück zum Zitat Groß, J.: Nonnegative-definite and positive-definite solution to the matrix equation AXA ∗=B—revisited. Linear Algebra Appl. 321, 123–129 (2000) MathSciNetMATHCrossRef Groß, J.: Nonnegative-definite and positive-definite solution to the matrix equation AXA =B—revisited. Linear Algebra Appl. 321, 123–129 (2000) MathSciNetMATHCrossRef
8.
Zurück zum Zitat Huang, L., Zeng, Q.: The solvability of matrix equation AXB+CYD=E over a simple Artinian ring. Linear Multilinear Algebra 38, 225–232 (1995) MathSciNetMATHCrossRef Huang, L., Zeng, Q.: The solvability of matrix equation AXB+CYD=E over a simple Artinian ring. Linear Multilinear Algebra 38, 225–232 (1995) MathSciNetMATHCrossRef
10.
Zurück zum Zitat Khatri, C.G., Mitra, S.K.: Hermitian and nonnegative definite solutions of linear matrix equations. SIAM J. Appl. Math. 31, 579–585 (1976) MathSciNetMATHCrossRef Khatri, C.G., Mitra, S.K.: Hermitian and nonnegative definite solutions of linear matrix equations. SIAM J. Appl. Math. 31, 579–585 (1976) MathSciNetMATHCrossRef
11.
12.
Zurück zum Zitat Liao, A.P., Bai, Z.Z., Lei, Y.: Best approximate solution of matrix equation AXB+CYD=E. SIAM J. Matrix Anal. Appl. 27, 675–688 (2006) MathSciNetCrossRef Liao, A.P., Bai, Z.Z., Lei, Y.: Best approximate solution of matrix equation AXB+CYD=E. SIAM J. Matrix Anal. Appl. 27, 675–688 (2006) MathSciNetCrossRef
13.
Zurück zum Zitat Liu, Y., Tian, Y.: More on extremal ranks of the matrix expressions A−BX±X ∗ B ∗ with statistical applications. Numer. Linear Algebra Appl. 15, 307–325 (2008) MathSciNetMATHCrossRef Liu, Y., Tian, Y.: More on extremal ranks of the matrix expressions ABX±X B with statistical applications. Numer. Linear Algebra Appl. 15, 307–325 (2008) MathSciNetMATHCrossRef
14.
Zurück zum Zitat Liu, Y., Tian, Y.: Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA ∗=B with applications. J. Appl. Math. Comput. 32, 289–301 (2010) MathSciNetMATHCrossRef Liu, Y., Tian, Y.: Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA =B with applications. J. Appl. Math. Comput. 32, 289–301 (2010) MathSciNetMATHCrossRef
15.
Zurück zum Zitat Liu, Y., Tian, Y., Takane, Y.: Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA ∗=B. Linear Algebra Appl. 431, 2359–2372 (2009) MathSciNetMATHCrossRef Liu, Y., Tian, Y., Takane, Y.: Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA =B. Linear Algebra Appl. 431, 2359–2372 (2009) MathSciNetMATHCrossRef
17.
Zurück zum Zitat Mitra, S.K.: A pair of simultaneous linear matrix equations A 1 XB 1=C 1, A 2 XB 2=C 2 and a programming problem. Linear Algebra Appl. 131, 107–123 (1990) MathSciNetMATHCrossRef Mitra, S.K.: A pair of simultaneous linear matrix equations A 1 XB 1=C 1, A 2 XB 2=C 2 and a programming problem. Linear Algebra Appl. 131, 107–123 (1990) MathSciNetMATHCrossRef
18.
Zurück zum Zitat Özgüle, A.B.: The matrix equation AXB+CYD=E over a principal ideal domain. SIAM J. Matrix Anal. Appl. 12, 581–591 (1991) MathSciNetCrossRef Özgüle, A.B.: The matrix equation AXB+CYD=E over a principal ideal domain. SIAM J. Matrix Anal. Appl. 12, 581–591 (1991) MathSciNetCrossRef
19.
Zurück zum Zitat Rosen, J.B.: Minimum and basic solutions to singular linear systems. J. Soc. Ind. Appl. Math. 12, 156–162 (1964) MATHCrossRef Rosen, J.B.: Minimum and basic solutions to singular linear systems. J. Soc. Ind. Appl. Math. 12, 156–162 (1964) MATHCrossRef
20.
Zurück zum Zitat Tian, Y., Liu, Y.: Extremal ranks of some symmetric matrix expressions with applications. SIAM J. Matrix Anal. Appl. 28, 890–905 (2006) MathSciNetMATHCrossRef Tian, Y., Liu, Y.: Extremal ranks of some symmetric matrix expressions with applications. SIAM J. Matrix Anal. Appl. 28, 890–905 (2006) MathSciNetMATHCrossRef
21.
Zurück zum Zitat Tian, Y., Tian, Y.: The minimal rank of the matrix expression A−BX−YC. Mo. J. Math. Sci. 14, 40–48 (2002) Tian, Y., Tian, Y.: The minimal rank of the matrix expression ABXYC. Mo. J. Math. Sci. 14, 40–48 (2002)
22.
Zurück zum Zitat Wang, Q.W., Jiang, J.: Extreme ranks of (skew-)Hermitian solutions to a quaternion matrix equation. Electron. J. Linear Algebra 20, 552–573 (2010) MathSciNetMATH Wang, Q.W., Jiang, J.: Extreme ranks of (skew-)Hermitian solutions to a quaternion matrix equation. Electron. J. Linear Algebra 20, 552–573 (2010) MathSciNetMATH
23.
Zurück zum Zitat Wang, Q.W., Zhang, F.: The reflexive re-nonnegative definite solution to a quaternion matrix equation. Electron. J. Linear Algebra 17, 88–101 (2008) MathSciNetMATH Wang, Q.W., Zhang, F.: The reflexive re-nonnegative definite solution to a quaternion matrix equation. Electron. J. Linear Algebra 17, 88–101 (2008) MathSciNetMATH
24.
Zurück zum Zitat Wang, Q.W., Sun, J.H., Li, S.Z.: Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra. Linear Algebra Appl. 353, 169–182 (2002) MathSciNetMATHCrossRef Wang, Q.W., Sun, J.H., Li, S.Z.: Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra. Linear Algebra Appl. 353, 169–182 (2002) MathSciNetMATHCrossRef
25.
Zurück zum Zitat Wang, Q.W., Song, G.J., Lin, C.Y.: Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application. Appl. Math. Comput. 189, 1517–1532 (2007) MathSciNetMATHCrossRef Wang, Q.W., Song, G.J., Lin, C.Y.: Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application. Appl. Math. Comput. 189, 1517–1532 (2007) MathSciNetMATHCrossRef
26.
Zurück zum Zitat Wang, Q.W., Zhang, H.S., Yu, S.W.: On solution to the quaternion matrix equation AXB+CYD=E. Electron. J. Linear Algebra 17, 343–358 (2008) MathSciNetMATH Wang, Q.W., Zhang, H.S., Yu, S.W.: On solution to the quaternion matrix equation AXB+CYD=E. Electron. J. Linear Algebra 17, 343–358 (2008) MathSciNetMATH
27.
Zurück zum Zitat Wei, M., Wang, Q.: On rank-constrained Hermitian nonnegative-definite least squares solutions to the matrix equation AXA ∗=B. Int. J. Comput. Math. 84, 945–952 (2007) MathSciNetMATHCrossRef Wei, M., Wang, Q.: On rank-constrained Hermitian nonnegative-definite least squares solutions to the matrix equation AXA =B. Int. J. Comput. Math. 84, 945–952 (2007) MathSciNetMATHCrossRef
29.
Zurück zum Zitat Zhang, X.: The general Hermitian nonnegative-definite solution to the matrix equation AXA ∗+BYB ∗=C. Acta Math. Acad. Paedagog. 21, 33–42 (2005) MATH Zhang, X.: The general Hermitian nonnegative-definite solution to the matrix equation AXA +BYB =C. Acta Math. Acad. Paedagog. 21, 33–42 (2005) MATH
30.
Zurück zum Zitat Zhang, X., Cheng, M.: The rank-constrained Hermitian nonnegative-definite and positive-definite solutions to the matrix equation AXA ∗=B. Linear Algebra Appl. 370, 163–174 (2003) MathSciNetMATHCrossRef Zhang, X., Cheng, M.: The rank-constrained Hermitian nonnegative-definite and positive-definite solutions to the matrix equation AXA =B. Linear Algebra Appl. 370, 163–174 (2003) MathSciNetMATHCrossRef
Metadaten
Titel
Ranks of submatrices in (skew-)Hermitian solutions to a quaternion matrix equation
verfasst von
Ning Li
Jing Jiang
Guang-Jing Song
Publikationsdatum
01.07.2013
Verlag
Springer-Verlag
Erschienen in
Journal of Applied Mathematics and Computing / Ausgabe 1-2/2013
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-012-0616-2

Weitere Artikel der Ausgabe 1-2/2013

Journal of Applied Mathematics and Computing 1-2/2013 Zur Ausgabe