Inverse of invertible standard multi-companion matrices with applications

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 multicompanion 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.