Stability of a matrix polynomial in continuous systems

Two sufficient conditions under which the roots of the determinant of a given (
m \times m
) matrix polynomial of
n
th order lie in the open left-half plane have been obtained. The first condition is given in terms of the positive definiteness of an (
mn \times mn
) symmetric matrix, while the second condition is given in terms of the positive definiteness of an (
m \times m
) matrix that is a function of
s
, Re
s \leq 0
. These conditions are represented in terms of rational functions of the coefficient matrices of the given matrix polynomial. Therefore, the explicit computation of the determinant polynomial is not required.