Inverse of invertible standard multi-companion matrices with applications

  • Authors

    • Hazem I. El Shekh Ahmed
    • Raid B. Salha
    • Diab I. AL-Awar
    2015-01-31
    https://doi.org/10.14419/ijamr.v4i1.4037
  • Standard multi-companion matrix, Inverse of invertible standard multi-companion matrix, Factorization, Jordan decomposition, Generalized eigenvectors.

  • Abstract

    The inverse of invertible standard multi-companion matrices will be derived and introduced as a new technique for generation of periodic autoregression models to get the desired spectrum and extract the parameters of the model from it when the information of the standard multi-companion matrices is not enough for the extracting of the parameters of the model.

    We will find explicit expressions for the generalized eigenvectors of the inverse of invertible standard multi-companion matrices such that each generalized eigenvector depends on the corresponding eigenvalue therefore we obtain a parameterization of the inverse of invertible standard multi-companion matrix through the eigenvalues and these additional quantities. The results can be applied to statistical estimation, simulation and theoretical studies of periodically correlated and multivariate time series in both discrete and continuous-time series.

  • References

    1. [1] B. Iqelan, "Periodically Correlated Time Series : Models and Example", LAP Lambert Academic Publishing, (2011).

      [2] G. Boshnakov, "Linear Algebra and its Applications", (2002).

      [3] G. Boshnakov, "Periodically correlated solutions to a class of stochastic difference equations", Stochastic differential and difference equations, (1997), 1-9.

      [4] I. Gohberg, P. Lancaster, L. Rodman, "Matrix polynomials", Computer Science and Applied Mathematics, (1982).

      [5] K. Hipel, A. McLeod, "Time series modelling of water resources and environmental systems", Developments in water science, (1994).

      [6] U. Kuechler, M. Soerensen, "A note on limit theorems for multivariate martingales", Bernoulli, 5, 3, (1999), 483-493.

      [7] P. Lancaster, L. Rodman, "Algebraic Riccati Equations", Oxford University Press, (1995).

      [8] P. Lancaster, M. Tismenetsky, "The Theory of Matrices", Academic Press, New York, (1985).

  • Downloads

  • How to Cite

    Ahmed, H. I. E. S., Salha, R. B., & AL-Awar, D. I. (2015). Inverse of invertible standard multi-companion matrices with applications. International Journal of Applied Mathematical Research, 4(1), 153-162. https://doi.org/10.14419/ijamr.v4i1.4037

    Received date: 2014-12-17

    Accepted date: 2015-01-12

    Published date: 2015-01-31