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(ε) = J + εB + O(ε²), where ε is real and positive. A necessary condition on B for the stability of A(ε) on an interval (0, ε₀), and a sufficient condition on B for the existence of such a family A(ε), is (i) Re trB ≤ 0; (ii) the sum of the elements on the first 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 definition based on the spectral radius. A generalized necessary condition, though not a sufficient condition, applies to arbitrary Jordan structures. The proof of our results uses two important techniques: the Puiseux-Newton diagram and the Arnold normal form. In the non-deragatory case our main results were obtained by Levantovskii in 1980 using a different proof. Practical implications are discussed.