Abstract
According to a result of Wigner and von Neumann, the dimension of the set \(\mathcal{M}\) of n × 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. Bibliography: 4 titles.
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References
P. Lax, Linear Algebra, Wiley, New York (1997).
Kh. D. Ikramov, “On the dimension of the variety of symmetric matrices with multiple eigenvalues, ” Zh. Vychisl. Matem. Matem. Fiz., 44, 963–967 (2004).
N. V. Ilyushechkin, “Discriminant of the characteristic polynomial of a normal matrix,” Matem. Zametki, 51, 16–23 (1992).
B. N. Parlett, “The (matrix) discriminant as a determinant,” Linear Algebra Appl., 355, 85–101 (2002).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 34–46.
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Dana, M., Ikramov, K.D. On the codimension of the variety of symmetric matrices with multiple eigenvalues. J Math Sci 137, 4780–4786 (2006). https://doi.org/10.1007/s10958-006-0275-7
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DOI: https://doi.org/10.1007/s10958-006-0275-7