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On the codimension of the variety of symmetric matrices with multiple eigenvalues. (Russian, English) Zbl 1081.15526

Zap. Nauchn. Semin. POMI 323, 34-46 (2005); translation in J. Math. Sci., New York 137, No. 3, 4780-4786 (2007).
Summary: According to a result of Wigner and von Neumann, the dimension of the set \({\mathcal M}\) of \(n\times n\) real symmetric matrices with multiple eigenvalues is equal to \(N- 2\), where \(N= n(n+ 1)/2\). This value is determined by counting the number of free parameters in the spectral decomposition of a matrix. We show that the same dimension is obtained if \({\mathcal M}\) is interpreted as an algebraic variety.

MSC:

15A30 Algebraic systems of matrices
15A18 Eigenvalues, singular values, and eigenvectors
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