I explain the univariate case first. My question is about generalizing this to multivariate cases.
Univariate Case Consider the following polynomial:
$$f(L)=1-\phi_1L-\phi_2L^2-\ldots-\phi_pL^p$$
Assume that the roots of this polynomial are denoted by ($L_i$ for $i=1...p$) and might be smaller or larger than one (in or outside the unit circle).
Now, assume that we calculate the reciprocal of those roots that are inside the unit circle and create a new set of roots. It is easy to calculate the coefficients of a polynomial, by using these new roots (e.g., see this answer).
(Please note that the application is related to estimating ARMA models).
Multivariate case
Consider the following polynomial matrix:
$$f(L)=\mathbf{I}_n-\boldsymbol{\Phi}_1L-\boldsymbol{\Phi}_2L^2-\ldots-\boldsymbol{\Phi}_pL^p$$
Similar to the univariate case, some roots of this polynomial are in the unit circle and others are not. So the question is, is it possible to generalize the univariate approach and regenerate a polynomial matrix for which all roots are outside the unit circle?
I know that the eigenvalues of the following matrix:
$$\mathbf{F}=\begin{bmatrix} \boldsymbol{\Phi}_1 & \boldsymbol{\Phi}_2 & \boldsymbol{\Phi}_3& \dots & \boldsymbol{\Phi}_{p-1}& \boldsymbol{\Phi}_p\\ \mathbf{I}_n& \mathbf{0}&\mathbf{0}&\ldots&\mathbf{0}&\mathbf{0}\\ \mathbf{0}& \mathbf{I}_n&\mathbf{0}&\ldots&\mathbf{0}&\mathbf{0}\\ \vdots& \vdots&\vdots&\ldots&\vdots&\vdots\\ \mathbf{0}& \mathbf{0}&\mathbf{0}&\ldots&\mathbf{I}_n&\mathbf{0}\\ \end{bmatrix} : np\times np $$
(with $\boldsymbol{\Phi}_i : n \times n$), satisfy:
$$ |\mathbf{I}_n\lambda^p-\boldsymbol{\Phi}_1\lambda^{p-1}-\boldsymbol{\Phi}_2\lambda^{p-2}-\ldots-\boldsymbol{\Phi}_p|=0 $$
(e.g., See Hamilton (1994) p. 259)
Therefore, I think maybe I can use the eigenvalues and eigenvectors of such matrix and calculate the polynomial. However, it is not clear how.
Thanks.
