TY - JOUR

T1 - Stable perturbations of nonsymmetric matrices

AU - Burke, James V.

AU - Overton, Michael L.

N1 - Funding Information:
We thank V. 1. Arnold fbr bringing the work ofL evantovskiit o our attention. We also thank R. S. Womersleyf i some helpful discussionsd uring the early stageso f this work. This research was supported in part by National Science Foundatiung rants DMS-9102059 and CCR-9101640.

PY - 1992/7/1

Y1 - 1992/7/1

N2 - 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.

AB - 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.

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U2 - 10.1016/0024-3795(92)90263-A

DO - 10.1016/0024-3795(92)90263-A

M3 - Article

AN - SCOPUS:0000549399

SN - 0024-3795

VL - 171

SP - 249

EP - 273

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - C

ER -