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On the partition of real skew-symmetric \(n\times n\)-matrices according to the multiplicities of their eigenvalues. (English) Zbl 0971.58005

Butzer, P. L. (ed.) et al., Karl der Große und sein Nachwirken. 1200 Jahre Kultur und Wissenschaft in Europa. Band 2: Mathematisches Wissen. Turnhout: Brepols. 439-454 (1998).
The main result of this paper is the following: Let \(A_n\) be the space of all skew-symmetric real \(n\times n\) matrices. Given a tuple \((m_0,\dots,m_k)\) of integers with \(m_0\geq 0\), \(m_1,\dots, m_k\geq 1\) and with \(n=m_0+2m_1+\cdots 2m_k\), let \(A_{m_0,m_1,\dots,m_k}\) be the set of all skew-symmetric matrices such that \(0\) is an \(m_0\)-fold eigenvalue, and which have \(2r\) distinct other eigenvalues in pairs \((i\lambda_r,-i\lambda_r)\) with multiplicity \(m_r\). Then the \(A_{m_0,\dots,m_k}\) form a Whitney regular stratification of \(A_n\), and the codimension of the stratum \(A_{m_0,\dots,m_k}\) is \(m_0(m_0-1) + \sum_{r=1}^k (m_r^2-1)\). Thom’s transversality theorem is applied to this theorem to get information about generic parameter dependent families of skew-adjoint matrices, with applications to control theory.
For the entire collection see [Zbl 0942.00019].

MSC:

58A35 Stratified sets
57Q65 General position and transversality
65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems
93B05 Controllability
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