About the uniqueness solution of the matrix polynomial equation A(λ)X(λ) − Y(λ)B(λ) = C(λ)

  • V. M. Prokip
Article

Abstract

We establish conditions for the existence of a unique solution of the matrix polynomial equation A(λ)X(λ) − Y(λ)B(λ) = C(λ) over an arbitrary field.

Key words and phrases

Polynomial matrix Linear equation Unique solution 

2000 Mathematics Subject Classification

15A06 15A22 15A24 93B25 

References

  1. 1.
    F. R. Gantmacher, Theory of Matrices (Chelsea, New York, 1964).Google Scholar
  2. 2.
    P. Lancaster, Theory of Matrices (Academic Press, New York, 1969).MATHGoogle Scholar
  3. 3.
    W. E. Roth, The Equation AX − XB = C and AX − Y B = C in Matrices, Proc. Amer.Math. Soc. 3, 392 (1952).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    S. Barnett, Regular Polynomial Matrices Having Relatively Prime Determinants, Proc. Camb. Phil. Soc. 65, 585 (1969).MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    H. Flanders and H. K. Wimmer, On the Matrix Equations AX − XB = C and AX − YB = C, SIAMJ. Appl.Math. 32, 707 (1977).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    W. H. Gustafson, Roth’s Theorem over Commutative Rings, Linear Algebra and Appl. 23, 245 (1979).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    J. Feinstein and Y. Bar-Ness, On the Uniqueness Minimal Solution of the Matrix Polynomial Equation A(λ)X(λ) + Y (λ)B(λ) = C(λ), J. Franklin Inst. 310, 131 (1980).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. K. Baksalary and R. Kala, TheMatrix Equation AX − Y B = C, Linear Algebra and Appl. 25, 41 (1979).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    R. Guralnick, Roth’s Theorems for Sets of Matrices, Linear Algebra and Appl. 71, 113 (1985).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    V. Olshevsky, Similarity of Block Diagonal and Block Triangular Matrices, Integral Equations and Operator Theory 15, 853 (1992).MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A. J. Ward and F. Gerrish, A Constructive Proof by Elementary Transformations of Roth’s Removal Theorems in the Theory of Matrix Equations, Int. J. Math. Educ. Technol. 31, 425 (2000).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    S. D. Brierley and E. B. Lee, Solution of the Equation A(z)X(z) − Y (z)B(z) = C(z) and Application to the Stability of Generalized Linear Systems, Int. J. Control. 40, 1965 (1984).CrossRefMathSciNetGoogle Scholar
  13. 13.
    T. Kaczorek, Zero-degree Solutions to the Bilateral Matrix Equations, Bull. Pol. Acad. Sci., Tech. Sci. 34, 547 (1986).MATHMathSciNetGoogle Scholar
  14. 14.
    V. M. Petrichkovich, Cell-triangular and cell-diagonal Factorizations of Cell-triangular and Celldiagonal Polynomial Matrices, Math. Notes. 37, 431 (1985).MATHMathSciNetGoogle Scholar
  15. 15.
    V. M. Prokip, On Divisibility and One-sided Equivalence of Polynomial Matrices, Ukr. Math. Zh. 42, 1213 (1990).MATHMathSciNetGoogle Scholar

Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  • V. M. Prokip

There are no affiliations available

Personalised recommendations