Primitivity of Positive Matrix Pairs: Algebraic Characterization, Graph Theoretic Description, and 2D Systems Interpretation

Abstract

In this paper the primitivity of a positive matrix pair (A,B) is introduced as a strict positivity constraint on the asymptotic behavior of the associated two-dimensional (2D) state model. The state evolution is first considered under the assumption of periodic initial conditions. In this case the system evolves according to a one-dimensional (1D) state updating equation, described by a block circulant matrix. Strict positivity of the asymptotic dynamics is equivalent to the primitivity of the circulant matrix, a property that can be restated as a set of conditions on the spectra of $A + e^{i \omega} B$, for suitable real values of $\omega$. The theory developed in this context provides a foundation whose analytical ideas may be generalized to nonperiodic initial conditions. To this purpose the spectral radius and the maximal modulus eigenvalues of the matrices $e^{i \theta} A + e^{i \omega} B$, $\theta$ and $\omega \in \hbox{{\bbb R}},$ are related to the characteristic polynomial of the pair (A,B) as well as to the structure of the graphs associated with A and B and to the factorization properties of suitable integer matrices. A general description of primitive positive matrix pairs is finally derived, including both spectral and combinatorial conditions on the pair.