Stable Perturbations of Nonsymmetric Matrices

A complex matrix is said to be stable if all its eigenvalues have negative real part. Let J be a Jordan block with zeros on the diagonal and ones on the superdiagonal, and consider analytic matrix perturbations of the form A() = J + B + O(2), where is real and positive. A necessary condition on B for the stability of A() on an interval (0; 0), and a suucient condition on B for the existence of such a family A(), is (i) Re tr B 0; (ii) the sum of the elements on the rst subdiagonal of B has nonpositive real part and zero imaginary part; (iii) the sum of the elements on each of the other subdiagonals of B is zero. This result is extended to matrices with any number of nonderogatory eigenvalues on the imaginary axis, and to a stability deenition based on the spectral radius. A generalized necessary condition, though not a suucient condition, applies to arbitrary Jordan structures. The proof of our results uses two important techniques: the Puiseux-Newton diagram and the normal form of Arnold. In the nonderoga-tory case our main results were obtained by Levantovskii in 1980 using a diierent proof. Practical implications are discussed.