Multi-companion matrices Original Research Article

In this paper, we introduce and study the class of multi-companion matrices. They generalize companion matrices in various ways and possess a number of interesting properties. We find explicit expressions for the generalized eigenvectors of multi-companion matrices such that each generalized eigenvector depends on the corresponding eigenvalue and a number of quantities which are functionally independent of the eigenvalues of the matrix and (up to a uniqueness constraint) of each other. Moreover, we obtain a parameterization of a multi-companion matrix through the eigenvalues and these additional quantities. The number of parameters in this parameterization is equal to the number of non-trivial elements of the multi-companion matrix. 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.