Stability of a matrix polynomial in discrete systems

Two sufficient conditions that the determinant of a nonsingular real (
m \times m
) matrix polynomial of
n
th order has all its roots inside the unit circle have been obtained. These conditions are represented in terms of rational functions of the coefficient matrices. Therefore, these conditions do not require the computation of the determinant polynomial. The first condition is given in terms of the positive definiteness of an (
mn \times mn
) symmetric matrix, while the second condition is expressed by the positive definiteness of an (
m \times m
) Hermitian matrix which is a function of
z, |z| \leq 1
. The first condition implies the second, and hence is more restrictive than the second.