Multi-companion matrices

Inverse of invertible standard multi-companion matrices with applications

“…, ..., x (s1) } and C 2 = {y (1) , y (2) , ..., y (s2) } be the two chains of generalized eigenvectors corresponding to x…”

“…its spectrum. In the general case, d > 1, the polynomials φ t (z) = 1 − Σφ t,i z i cannot be used with the same success but a natural generalization exists (see [2]). Let m = max(d, p 1 , ..., p d ), Z t = (X t , X t−1 , ..., X t−m+1 ).…”

“…Equation (10) can be used to give full description of the properties of the periodic autoregressive process {X t }, for more see [2].…”

Generation Of Time Series Models With Given Spectral Properties

“…The main aim of this article is to give a method for generation of periodically correlated and multivariate ARIMA models whose dynamic characteristics are partially or fully specified in terms of spectral poles and zeroes or their equivalents in the form of eigenvalues/eigenvectors of the associated model matrices. Our method uses a reparameterization of the models based on the spectral decomposition of multi‐companion matrices and their factorization into products of companion matrices developed by Boshnakov (2002). We are not aware of any general method for multivariate linear systems of comparable generality and control over the spectral properties of the generated model.…”

“…In , we present the necessary material about multi‐companion matrices (spectral properties and companion factorization). It is based on Boshnakov (2002) with some additions. discusses how the theory applies to multivariate and periodically correlated ARIMA‐type models.…”