Skip to main content
SpringerLink
Log in

Solutions of the matrix equation (1):X m+A 1Xm−1+…+A m=0

  • Published:
Acta Mathematica Sinica Aims and scope Submit manuscript

Abstract

SupposeX and the coefficientsA 1, …,A m aren×n matrices. LetB be anmn×mn matrix as in (7). LetJ be the Jordan canonical matrix ofB andB=PJP −. LetE i denote thei×i unit matrix.V is defined bydV/dt=JV andV(t=0)=E mn. ThenZ=PV satisfiesdZ/dt=BZ.P * is a matrix which consists of the firstn rows ofP. The author proves: There is a solution of (1) ↔ there are anmn×n matrixC, ann×n matrixQ and ann×n function matrixN such thatP *VC=QN, where detQ≠0 andN is defined byN(t=0)=E n anddN/dt=RN, in whichR is ann×n Jordan canonical matrix. There are three cases regarding the solutions of (1): No solution, finitek solutions, 1<k<C mn , and infinite solutions which containj parameters, 1<-j<-mn 2.

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. Thurston H.S. and Mary K. Alexander,The equation X 2+PX+Q=0 in binary matrices, Amer. Math. Monthly, (1954), 8–17.

  2. Flanders, Harley,On polynomial matrix equations, College Math. J.,17 (1986), 388–391.

    Article  MATH  MathSciNet  Google Scholar 

  3. Petkov, Milko,On the matrix equation A 0Xm+A1Xm−1+…+Am=0, (A m0λ +A m−10λ +…+Am)x=0, Bulgarian, English summary, Annuaire Univ. Sofia Fac. Math. Mec.,72 (1978), 159–164 (1986).

    MathSciNet  Google Scholar 

  4. Maloulas, J.,A theorem on the factorization of matrix polynomials, Linear Algebra Appl.,84 (1986), 311–322.

    Article  MathSciNet  Google Scholar 

  5. Mazko, A.G.,Distribution of a matrix polynomial with respect to plane curves, Russian, Numerical-Analytical Math., 90–96, Akad Nauk Ukrain SSR, Inst. Mat. Kiev, 1984.

  6. King, Feng; Wang, Kai,On the g-circulant solutions to the matrix equation A m=λJ, II. J. Combin. Theory Ser. A38 (1985), 182–186.

    Article  MATH  MathSciNet  Google Scholar 

  7. Petrichkovich, V.M.,Linear divisors and reducibility of polynomial matrices Russian, Ukrain Mat. Zh.,36 (1984), 195–200.

    MathSciNet  Google Scholar 

  8. Zelisko, V.R.,Questions of factorization of matrix polynomials, Mat. Metody i Fiz-Mekh Polya, no.17 (1983), 28–33.

    MATH  MathSciNet  Google Scholar 

  9. Cohen, Nir.,Spectral analysis of regular matrix polynomials, Integral Equations Operator Theory,6 (1983), 161–183.

    Article  MATH  MathSciNet  Google Scholar 

  10. Gracia J.M., I. Zaballa and M.A. Beitia, Algebra linealy applications, Proceedings of the Conference held in Victoria-Gasteiz, September 28–30, 1983, Univ. del Pais Vasco-Euskal Herriko Unibertsitatia Bibao, 1984.

  11. Hernandez Garcia, Vicente; Incertis Carro, Fernando,Solution of polynomial matrix equations by means of annihilation polynomials of a matrix solution, Rev. Real Akad. Science Exact. Fis. Natur. Madrid,26 (1982), 131–154.

    Google Scholar 

  12. Gohberg, I., Lancaster, P., Rodman, L., Matrix polynomials, Computer Science and Applied Mathematics, Academic Press, Inc, New York-London, 1982, xiv+409pp.

    MATH  Google Scholar 

  13. Gohberg, I., Kaashoek, M.A., Lerer, L., Rodman, L.,Common multiples and common divisors of matrix polynomials II. Vandermonde and resultant matrices, Linear and Multilinear Algebra,12 (1982/1983), 159–203.

    MathSciNet  Google Scholar 

  14. Wang, Kai,On the g-circulant solutions to matrix equation A m=λJ, J. Combin. Theory Ser. A,33, (1982), 287–296.

    Article  MathSciNet  Google Scholar 

  15. Chenggui Huang, The solutions of matrix equation, 1982, p. 191, Abstracts of the AMS.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tianjin Normal University, PRC

    Huang Cheng-gui

Authors
  1. Huang Cheng-gui
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cheng-gui, H. Solutions of the matrix equation (1):X m+A 1Xm−1+…+A m=0. Acta Mathematica Sinica 8, 225–235 (1992). https://doi.org/10.1007/BF02582911

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02582911

Keywords

Search

Navigation