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A new condition number of the eigenvalue and its application in control theory. (English) Zbl 0665.15006

Given a real \(n\times n\) matrix A and its standardized eigenvectors or generalized eigenvectors \(x_ i\), \(i=1,...,n\), with \(X\triangleq (x_ 1,...,x_ n)\), \(\Delta =\det (X^ TX)\), the author defines the new condition number \(K(A)\triangleq 1/\Delta\). To compare it with \(K(A)=\sigma_ 1/\sigma_ n\), where \(\sigma_ 1\), \(\sigma_ n\) are the greatest and the smallest singular values of X, he proves that \(\sigma^ 2_ n\geq 1/[1+(n/n-1)^{n-1}/\Delta]\). According to this evaluation he proposes the following nonlinear programming problem: max det(X\({}^ TX)\) subject to \((A+BK)X=X\Lambda\), \(x^ T_ ix_ i=1\), \(i=1,...,n\), with B, K matrices of adequate dimensions and \(\Lambda \triangleq diag(\lambda_ 1,...,\lambda_ n)\), as a associated one to the robust pole assignment of \(A+BK\) via K and with the prescribed eigenvalues \(\lambda_ 1,...,\lambda_ n\). Two examples illustrate this latter problem.
Reviewer: M.Voicu

MSC:

15A12 Conditioning of matrices
93B55 Pole and zero placement problems
15A42 Inequalities involving eigenvalues and eigenvectors
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