access icon free On coprimeness of two polynomials in the framework of conjugate product

The right and left con-Sylvester matrices for two polynomials are defined, and then the criteria for right and left coprimeness of two polynomials in the framework of conjugate product are given in terms of the determinant of the right and left con-Sylvester matrices, respectively. The results in this study can be viewed as a generalisation of the well-known Sylvester resultant criterion for coprimeness of two polynomials in the framework of ordinary product.

Inspec keywords: polynomial matrices

Other keywords: ordinary product; coprimeness; Sylvester resultant criterion; conjugate product; con-Sylvester matrices; polynomials

Subjects: Algebra; Algebra, set theory, and graph theory; Algebra; Algebra

References

    1. 1)
      • 7. Vogt, W.G., Bose, N.K.: ‘A method to determine whether two polynomials are relatively prime’, IEEE Trans. Autom. Control, 1970, AC-12, (3), pp. 379380.
    2. 2)
      • 9. Wu, A.G., Feng, G., Liu, W.Q., et al: ‘The complete solution to the Sylvester-polynomial-conjugate matrix equations’, Math. Comput. Model., 2011, 53, pp. 20442056.
    3. 3)
      • 10. Duan, G.R.: ‘Generalized Sylvester equations: unified parametric solutions’ (CRC Press, Boca Raton, 2015).
    4. 4)
      • 2. Barnett, S.: ‘Matrices, polynomials, and linear time-invariant systems’, IEEE Trans. Autom. Control, 1973, 18, (1), pp. 110.
    5. 5)
      • 14. Ferrante, A., Pavon, M., Ramponi, F.: ‘Hellinger versus Kullback–Leibler multivariable spectrum approximation’, IEEE Trans. Autom. Control, 2008, 53, (4), pp. 954967.
    6. 6)
      • 12. Wu, A.G., Qian, Y.Y., Liu, W.: ‘Linear quadratic regulation for discrete-time antilinear systems: an anti-Riccati matrix equation approach’, J. Franklin Inst., 2016, 353, (5), pp. 10411060.
    7. 7)
      • 5. Hajarian, M.: ‘Matrix GPBiCG algorithms for solving the general coupled matrix equations’, IET Control Theory Appl., 2015, 9, (1), pp. 7481.
    8. 8)
      • 13. Wu, A.G., Zhang, Y., Liu, W., et al: ‘State response for continuous-time antilinear systems’, IET Control Theory Appl., 2015, 9, (8), pp. 12381244.
    9. 9)
      • 6. Jury, E.I.: ‘Inners and stability of dynamic systems’ (Wiley, New York, 1974).
    10. 10)
      • 11. Wu, A.G., Duan, G.R., Feng, G., et al: ‘On conjugate product of complex polynomials’, Appl. Math. Lett., 2011, 24, pp. 735741.
    11. 11)
      • 3. Kailath, T.: ‘Linear systems’ (Prentice-Hall, Englewood Cliffs, 1980).
    12. 12)
      • 16. Wu, A.G., Liu, W., Duan, G.R.: ‘On the conjugate product of complex polynomial matrices’, Math. Comput. Model., 2011, 53, (9-10), pp. 20312043.
    13. 13)
      • 1. Kalman, R.E.: ‘Mathematical description of linear systems’, SIAM J. Control, 1963, 1, pp. 128151.
    14. 14)
      • 8. Zhou, B., Duan, G., Song, S.: ‘An identity concerning controllability observability and coprimeness of linear systems and its applications’, J. Control Theory Appl., 2007, 5, (2), pp. 177183.
    15. 15)
      • 15. Pavon, M., Ferrante, A.: ‘On the Georgiou–Lindquist approach to constrained Kullback–Leibler approximation of spectral densities’, IEEE Trans. Autom. Control, 2006, 51, (4), pp. 639644.
    16. 16)
      • 4. de Souza, E., Bhattacharyya, S.P.: ‘Controllability, observability and the solution of AXXB=C’, Linear Algebra Appl., 1981, 39, pp. 167188.
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