On matrix perturbations with minimal leading Jordan structure.
Let \(A(\varepsilon )=A_0+A_1\varepsilon +O(\varepsilon ^2)\) be a matrix perturbation of an \(n\times n\) nilpotent matrix \(A_0\) over the set of complex numbers. Assuming that the Jordan canonical form of the leading matrix \(A_0\) is known, \textit{V. B. Lidskij} [Perturbation theory of non-conjugate operators. U.S.S.R. Comput. Math. Math. Phys., 6, 73--85 (1966); transl. from Zh. Vychisl. Mat. Mat. Fiz. 6, No. 1, 52--60 (1966; Zbl 0166.40501)] provided sufficient conditions on some entries of \(A_1\) for \(A(\varepsilon )\) to admit \(rm\) eigenvalues with order \(O(\epsilon ^{1/m})\) when \(A_0\) has \(r\) nilpotent Jordan blocks with size \(m.\) However Lidskij's conditions are not necessary and the same eigenvalue splitting may occur for some other leading Jordan structures. The author proves that any matrix perturbation of an \(n\times n\) nilpotent complex matrix is similar to a matrix perturbation whose leading coefficient has minimal Jordan structure. In addition, the author derive the property that, for matrix perturbations with minimal leading Jordan structure, the sufficient conditions of Lidskij's theorem are also necessary. The main tools for deriving these results are the Newton diagram associated with the characteristic polynomial of a matrix perturbation and the Newton envelope associated with its leading matrix.
scientific article; zbMATH DE number 2032616
scientific article