Class of rational function matrices that satisfy two properties of linear systems and structural controllability

It is shown that the type-1 matrix satisfies two already introduced properties, namely, that the non-zero eigenvalues of A have multiplicity 1 for almost all zisinR^q, and that a non-zero constant r is not an eigenvalue of A for almost all zisinR^q. That is, the characteristic polynomial in ring F(z)[lambda] has no non-zero constant eigenvalues or non-zero multiple factors. Some controllability criteria of the linear systems, including the ones whose characteristic polynomials have no non-zero multiple factors in F(z)[lambda], are derived. The application of the type-1 matrix and system to structural controllability is shown.