Abstract.A perturbation result, due to R. Rado and presented by H. Perfect in 1955, shows how to modify r eigenvalues of a matrix of order n, r ≤ n, via a perturbation of rank r, without changing any of the n − r remaining eigenvalues. This result extended a previous one, due to Brauer, on perturbations of rank r = 1. Both results have been exploited in connection with the nonnegative inverse eigenvalue problem. In this paper a symmetric version of Rado's extension is given, which allows us to obtain a new, more general, sufficient condition for the existence of symmetric nonnegative matrices with prescribed spectrum.Key words. Symmetric nonnegative inverse eigenvalue problem.
A nonsingular configuration of m lines of ℝP3 is defined as an (unordered) collection of m nonoriented pairwise disjoint lines in ℝP3. An isotopy of such a collection in the process of which the lines remain pairwise disjoint lines is called a rigid isotopy. The main aim of the paper is to classify nonsingular configurations of 7 lines of ℝP3 up to rigid isotopy.
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