Cyclic dimensions, kernel multiplicities, and Gohberg-Kaashoek numbers
Let \(A\) be an \(n\times n\) complex matrix. Two quantities are defined: The \(d\)-th cyclic dimension \(r_d (A)\) is the maximal dimension over all \(A\)-invariant subspaces generated by \(d\) vectors. The \(d\)-th kernel multiplicity is the maximal dimension of the kernel of \(f(A)\) over all polynomials \(f(\lambda)\) whose degrees do not exceed \(d\). It is proved that the cyclic dimensions can only increase and the kernel multiplicities can only decrease under small (in the norm) perturbations of \(A\). The Gohberg-Kaashoek numbers \(m_i (A)\) are defined as follows: Let \(m_1 (A, \lambda)\geq m_2 (A, \lambda)\geq \cdots\) be the sizes of the Jordan blocks of \(A\) corresponding to its eigenvalue \(\lambda\); then \(m_i (A)= \sum_{\lambda\in \sigma (A)} m_i (A, \lambda)\). It is proved that \[ r_d (A)= \sum^d_{i=1} m_i (A), \] and a similar relation exists between the kernel multiplicities and the dual sequence of the Gohberg-Kaashoek numbers. Using these relationships, a geometric proof is given for a theorem by \textit{H. Den Boer} and \textit{G. Ph. A. Thijsse} [Integral Equations Oper. Theory 3, 23-42 (1980; Zbl 0433.47007)] and \textit{A. S. Markus} and \textit{E. E. Parilis} [Mat. Issled. 54, 98-109 (1980; Zbl 0453.15010)]; engl. transl. in Linear Algebra Appl. 54, 139-152 (1983; Zbl 0524.15010)] that asserts that the Gohberg-Kaashoek numbers of small perturbations of \(A\) majorize the Gohberg-Kaashoek numbers of \(A\), and that this majorization is the only restriction on the Jordan structure of small perturbations of \(A\).
scientific article; zbMATH DE number 881693