Mediterranean Journal of Mathematics

, Volume 9, Issue 1, pp 47–60 | Cite as

On Additive Decomposition of the Hermitian Solution of the Matrix Equation AXA* = B

  • Yongge Tian


The decomposition of a Hermitian solution of the linear matrix equation AXA* = B into the sum of Hermitian solutions of other two linear matrix equations \({A_{1}X_{1}A^{*}_{1} = B_{1}}\) and \({A_{2}X_{2}A^*_{2} = B_{2}}\) are approached. As applications, the additive decomposition of Hermitian generalized inverse C = A + B for three Hermitian matrices A, B and C is also considered.

Mathematics Subject Classification (2010)

15A03 15A09 15A24 15B57 


Matrix equation Hermitian solution decomposition of solution set inclusion generalized inverse rank formulas for partitioned matrices 


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  1. 1.
    Anderson W.N., Duffin R.J.: Series and parallel addition of matrices. J. Math. Anal. Appl. 26, 576–594 (1969)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Baksalary J.K.: Nonnegative definite and positive definite solutions to the matrix equation AXA* = B. Linear and Multilinear Algebra 16, 133–139 (1984)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Ben-Israel A., Greville T.N.E.: Generalized Inverses: Theory and Applications. 2nd Edition, Springer, New York (2003)MATHGoogle Scholar
  4. 4.
    Bernstein D.S.: Matrix Mathematics: Theory, Facts and Formulas. 2nd Edition, Princeton University Press, Princeton (2009)MATHGoogle Scholar
  5. 5.
    Dai H., Lancaster P.: Linear matrix equations from an inverse problem of vibration theory. Linear Algebra. Appl. 246, 31–47 (1996)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Eriksson-Bique S.-L., Leutwiler H.: A generalization of parallel addition. Aequationes Math. 38, 99–110 (1989)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Fill J.A., Fishkind D.E.: The Moore–Penrose generalized inverse for sums of matrices. SIAM J. Matrix Anal. Appl. 21, 629–635 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Groß J.: A note on the general Hermitian solution to AXA* = B. Bull. Malaysian Math. Soc. (2nd Ser.) 21, 57–62 (1998)MATHGoogle Scholar
  9. 9.
    Groß J.: Nonnegative-definite and positive-definite solutions to the matrix equation AXA* = B–revisited. Linear Algebra Appl. 321, 123–129 (2000)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hartwig R.E.: A remark on the characterization of the parallel sum of two matrices. Linear and Multilinear Algebra 22, 193–197 (1987)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    L. Hogben, Handbook of Linear Algebra. Chapman & Hall/CRC, 2007.Google Scholar
  12. 12.
    Khatri C.G., Mitra S.K.: Hermitian and nonnegative definite solutions of linear matrix equations. SIAM J. Appl. Math. 31, 579–585 (1976)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    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)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    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)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Y. Liu and Y. Tian, Max-min problems on the ranks and inertias of the matrix expressions ABXC ± (BXC)* with applications. J. Optim. Theory Appl., accepted.Google Scholar
  16. 16.
    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)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Marsaglia G., Styan G.P.H.: Equalities and inequalities for ranks of matrices. Linear and Multilinear Algebra 2, 269–292 (1974)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mitra S.K., Odell P.L.: On parallel summability of matrices. Linear Algebra Appl. 74, 239–255 (1986)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Mitra S.K., Prasad K.M.: The nonunique parallel sum. Linear Algebra Appl. 259, 77–99 (1997)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Mitra S.K., Prasad K.M.: The regular shorted matrix and the hybrid sum. Adv. Appl. Math. 18, 403–422 (1997)MATHCrossRefGoogle Scholar
  21. 21.
    Mitra S.K., Puri M.L.: On parallel sum and difference of matrices. J. Math. Anal. Appl. 44, 92–97 (1973)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Mitra S.K., Puri M.L.: Shorted matrices–an extended concept and some applications. Linear Algebra Appl. 42, 57–79 (1982)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Tian Y.: Equalities and inequalities for inertias of Hermitian matrices with applications. Linear Algebra Appl. 433, 263–296 (2010)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Y. Tian, On additive decompositions of solutions of the matrix equation AXB = C. Calcolo, doi: 10.1007/s10092-010-0019-4.
  25. 25.
    Tian Y., Liu Y.: Extremal ranks of some symmetric matrix expressions with applications. SIAM J. Matrix Anal. Appl. 28, 890–905 (2006)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Tian Y., Styan G.P.H.: On some matrix equalities for generalized inverses with applications. Linear Algebra Appl. 430, 2716–2733 (2009)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.China Economics and Management AcademyCentral University of Finance and EconomicsBeijingChina

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