Skip to main content
SpringerLink
Log in

The general Hermitian nonnegative-definite and positive-definite solutions to the matrix equationGXG *+HYH *=C

  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

A matrix pair (X 0,Y 0) is called a Hermitian nonnegative-definite (respectively, positive-definite) solution to the matrix equation

$$GXG* + HYH* = C$$

with unknownX andY ifX 0 andY 0 are Hermitian nonnegative-definite (respectively, positive-definite) and satisfyGX 0G*+HY0H*=C. Necessary and sufficient conditions for the existence of at least a Hermitian nonnegative-definite (respectively, positive-definite) solution to the matrix equation are investigated. A representation of the general Hermitian nonnegative-definite (respectively positive-definite) solution to the equation is also obtained when it has such solutions. Two presented examples show these advantages of the proposed approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. K. Baksalary and R. Kala,The matrix equation AXB+CYD=E, Linear Algebra Appl. 30:141–147(1980).

    Article  MATH  MathSciNet  Google Scholar 

  2. J. K. Baksalary,Nonnegative definite and positive definite solutions to the matrix equation AXA *=B, Linear and Multilinear Algebra 16:133–139(1984).

    Article  MATH  MathSciNet  Google Scholar 

  3. X. W. Chang and J. S. Wang,The symmetric solution of the matrix equations AX+XA=C, AXA T+BYBT=C, and (ATXa, BTXB)=(C, D), Linear Algebra Appl. 179:171–189(1993).

    Article  MATH  MathSciNet  Google Scholar 

  4. K. E. Chu,Singular value and generalized singular value decompositions and the solution of linear matrix equations, Linear Algebra Appl. 88/89:83–98(1987).

    Article  Google Scholar 

  5. E. V. Dulov,Algorithms for solving matrix polynomial equations of special form, J. Appl. Math. and Computing (old KJCAM) 7: 41–60(2000).

    MATH  MathSciNet  Google Scholar 

  6. J. Groß,Hermitian and nonnegative definite solutions of linear matrix equations, Bull. Malay. Math. Soc. 21:57–62(1998).

    MATH  Google Scholar 

  7. J. Groß,Nonnegative-define and positive solutions to the matrix equation AXA *=B—revisited, Linear Algebra Appl. 321:123–129(2000).

    Article  MATH  MathSciNet  Google Scholar 

  8. C. N. He,The general solution of the matrix equation AXB+CYD=F (in Chinese), Acta Sci. Nat. Univ. Norm. Hunan 19:17–20(1996).

    MATH  Google Scholar 

  9. C. G. Khatri and S. K. Mitra,Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. Appl. Math. 31:579–585 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  10. G. P. Xu, M. S. Wei and D. S. Zheng,On solutions of matrix equation AXB+CYD=F, Linear Algebra Appl. 279:93–109(1998).

    Article  MATH  MathSciNet  Google Scholar 

  11. J. H. Yun and S. W. Kim,Generalized stationary iterative method for solving linear systems, J. Appl. Math. and Computing(old KJCAM) 5: 341–349(1998).

    MATH  MathSciNet  Google Scholar 

  12. X. Zhang and M.-Y. Cheng,The rank-constrained Hermitian nonnegative-definite and positive-definite solutions to the matrix equation AXA *=B, Linear Algebra Appl., to appear.

  13. X. Zhang, S. Thompson and G.-R. Duan,Full-column Rank Solutions of the Matrix Equation AV-EVJ, Applied Mathmatics and Computation, to appear.

  14. X. Zhang and M.-Y. Cheng,The general common nonnegative-definite and positive-definite solutions to the matrix equation AXA *=BB* and CXC*=DD*, Applied Mathematics Letters, to appear.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Heilongjiang Univ., 150080, Harbin, P. R. China

    Xian Zhang

  2. School of Mechanical and Manufacturing Engineering, The Queen's University of Belfast, Stranmillis Road, BT9 5AH, Belfast, UK

    Xian Zhang

Authors
  1. Xian Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xian Zhang.

Additional information

This work was supported in part by the Chinese Natural Science Foundation under No. 10271021, the Natural Science Foundation of Heilongjiang Province under No. A01-07 and the NSF of Heilongjiang Education Committee under No. 15011014.

Xian Zhang received his BS from Heilongjiang University. Since 1990 he has been at the Heilongjiang University, which named him an Assistant in 1994. In September of 2001, he received a Assistant Professor from Heilongjiang Education Committee. In November of 2001, he come to UK for his P.h. D. degree. His research interests center on the theory of matrix algebra, the theory of linear control, and theirs applications.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, X. The general Hermitian nonnegative-definite and positive-definite solutions to the matrix equationGXG *+HYH *=C . JAMC 14, 51–67 (2004). https://doi.org/10.1007/BF02936098

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02936098

AMS Mathematics Subject Classification

Key words and phrases

Search

Navigation