Multi-companion matrices (Q92814)

Multi-companion matrices are defined, characterized, and studied. They generalize companion matrices. Generalized eigenvectors of multi-companion matrices are described and parametrized by their eigenvalues. It is found that each (generalized) eigenvector of a \(k\)-companion \(m\times m\) matrix can be represented as a simple function of the corresponding eigenvalue and \(k-1\) quantities, which are functionally independent from all eigenvalues of the matrix and the generalized eigenvectors corresponding to different eigenvalues. An application in periodic autoregression and in multivariate autoregression in continuous time is described.